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A Comparison of Optimization Algorithms for Deep Learning
Derya Soydaner
TL;DR
Deep-network optimization becomes more difficult as networks deepen and datasets grow, motivating alternatives to SGD. The paper examines adaptive algorithms across supervised and unsupervised tasks and four image datasets, finding that adaptive methods are mostly superior and computationally efficient for both tasks.
Problem
SGD faces ill-conditioning, large-dataset training costs, manual learning-rate tuning, and difficult parallelization in deep neural networks.
Method
The paper compares widely used adaptive optimization algorithms with SGD and momentum variants across supervised classification and unsupervised image-reconstruction tasks.
Results
Adaptive methods are mostly superior and computationally efficient, with better results for both supervised and unsupervised tasks.
Takeaways & Limitations
Adaptive learning algorithms remain important optimization choices for deep learning, while research on better adaptive methods continues.
Takeaways & Limitations
SGD with momentum requires an additional momentum hyperparameter, while AdaGrad can excessively reduce its effective learning rate and stop too early.
Abstract
from arXiv · showhide
In recent years, we have witnessed the rise of deep learning. Deep neural networks have proved their success in many areas. However, the optimization of these networks has become more difficult as neural networks going deeper and datasets becoming bigger. Therefore, more advanced optimization algorithms have been proposed over the past years. In this study, widely used optimization algorithms for deep learning are examined in detail. To this end, these algorithms called adaptive gradient methods are implemented for both supervised and unsupervised tasks. The behaviour of the algorithms during training and results on four image datasets, namely, MNIST, CIFAR-10, Kaggle Flowers and Labeled Faces in the Wild are compared by pointing out their differences against basic optimization algorithms.
1. Introduction
Deep-network optimization becomes harder with ill-conditioning and large datasets, motivating adaptive methods that adjust each parameter’s learning rate. The paper compares these methods with basic optimizers across supervised and unsupervised image-processing tasks.
- Motivation: SGD struggles with ill-conditioning, large-scale training, manual learning-rate tuning, and difficult parallelization.These limitations motivated more advanced optimization algorithms.
- Adaptive methods: Adaptive gradient methods adjust each parameter’s learning rate according to gradient magnitude.Large gradients receive reduced learning rates, while smaller gradients receive comparatively larger rates.
- Study aim: The study compares widely used adaptive optimization algorithms and their working principles in image processing.The comparison is framed against basic optimization algorithms.
- Evaluation design: CNNs are evaluated for image classification, while convolutional autoencoders and denoising autoencoders support unsupervised image reconstruction.Training behaviour and performance are examined across both task types.
2. Related Work
Related work studies optimization algorithms’ convergence, generalization, and numerical properties, while newer adaptive variants continue to extend the method family.
- Prior studies: Prior work analyzes convergence and generalization of RMSProp and Adam against Nesterov’s accelerated gradient method.Other comparisons include conjugate gradient, SGD, and limited-memory BFGS.
- Recent variants: Newer adaptive variants include AdaBound and AMSBound, which use dynamic learning-rate bounds to transition gradually toward SGD.Another method adapts learning rates both locally for each parameter and globally across parameters.
3. Optimization Algorithms with Adaptive Learning Rates
The paper reviews minibatch optimization, contrasting conventional SGD and momentum variants with methods that adapt learning rates during training. It explains their update mechanisms and practical trade-offs.
- Minibatch optimization: Minibatch methods process only part of the training set at each epoch and are widely preferred in deep learning.They lie between deterministic batch methods and one-example-at-a-time stochastic methods.
- Stochastic gradient descent: SGD estimates gradients from randomly selected minibatches and updates parameters using a learning rate that typically decreases over time.The learning rate strongly affects update magnitude and can be selected or adjusted through practical tuning.
- Momentum variants: Momentum addresses poor conditioning and stochastic-gradient variance by incorporating previous updates into the current change.This can accelerate convergence and reduce oscillation.
- Momentum variants: SGD with momentum uses a velocity variable representing the direction and speed of parameter movement.The velocity is an exponentially decaying average of negative gradients, controlled by momentum parameter α.
- Momentum variants: Momentum’s disadvantage is the additional requirement to tune a momentum hyperparameter alongside the learning rate.Common α values cited are 0.5, 0.9, and 0.99.
- Nesterov momentum: Nesterov momentum computes the gradient at an interim point after applying the current velocity direction.The algorithm performs an interim minibatch update before computing the velocity and final parameter update.
3.2. AdaGrad
AdaGrad adapts individual parameter learning rates using accumulated historical squared gradients, but this accumulation can reduce rates excessively during deep-network training.
- AdaGrad mechanism: AdaGrad individually adapts parameter learning rates using historical squared gradient values.Parameters with larger partial derivatives experience faster learning-rate decreases.
- AdaGrad update: AdaGrad initializes an accumulation variable r to zero before computing the minibatch gradient.The accumulated quantity is then used in subsequent updates.
- AdaGrad update: AdaGrad scales learning rates inversely with the square root of accumulated historical squared gradients.The global learning rate is ϵ, while δ provides numerical stability.
- Limitations: AdaGrad can stop too early because accumulated squared gradients excessively shrink the effective learning rate in deep neural networks.It performs well for some, but not all, deep-learning models.
3.3. AdaDelta
AdaDelta addresses AdaGrad’s decaying learning rates and manual global learning-rate selection by using a finite, exponentially weighted history of squared gradients. The resulting method requires no manual learning-rate tuning and is reported robust across several conditions.
- 3.3. AdaDelta: AdaDelta replaces AdaGrad’s unbounded accumulation of squared gradients with a fixed-size window of recent gradients.This prevents the learning rate from shrinking indefinitely as training progresses.
- 3.3. AdaDelta: The windowed accumulation is implemented as an exponentially decaying average, avoiding inefficient storage of w previous squared gradients.The decay constant ρ controls the running average.
- 3.3. AdaDelta: AdaDelta uses the root mean square of previous squared gradients when computing parameter updates.A small constant δ is included in the update computation.
- 3.3. AdaDelta: AdaDelta requires no manual learning-rate tuning and appears robust to noisy gradients, model architectures, data modalities, and hyperparameter choices.The robustness claim is reported across these varied conditions.
3.4. RMSProp
RMSProp modifies AdaGrad by replacing full-history gradient accumulation with an exponentially weighted moving average. The supplied passages also describe Adam’s related moment-based updates and practical advantages.
- 3.4. RMSProp: RMSProp replaces AdaGrad’s full-history squared-gradient accumulation with an exponentially decaying average.Discarding the distant past is intended to support rapid convergence after entering a convex bowl.
- 3.4. RMSProp: RMSProp accumulates the squared gradient after each gradient computation, using ρ as the decay rate, before computing and applying the parameter update.The passages introduce this as the implementation sequence for RMSProp.
- 3.4. RMSProp: Adam computes individual adaptive learning rates from first- and second-moment estimates of gradients, combining properties associated with AdaGrad and RMSProp.AdaGrad is described as effective with sparse gradients, while RMSProp is associated with online and non-stationary settings.
- 3.4. RMSProp: Adam updates exponential moving averages of gradients and squared gradients, then applies bias corrections before calculating parameter updates.The first- and second-moment variables are denoted m and u in the supplied passages.
- 3.4. RMSProp: Adam requires little learning-rate tuning, is computationally efficient, and is suitable for non-stationary objectives and noisy or sparse gradients.The passage also notes straightforward implementation, diagonal-rescaling invariance, and low memory requirements.
3.6. AdaMax
AdaMax extends Adam by replacing its L2-norm-based scaling with an infinity-norm formulation. It computes a biased first-moment estimate and exponentially weighted infinity norm before updating parameters.
- 3.6. AdaMax: AdaMax is an Adam extension based on the infinity norm.It generalizes Adam’s L2-norm-based scaling toward an Lp-norm formulation.
- 3.6. AdaMax: AdaMax computes gradients, a biased first-moment estimate, and an exponentially weighted infinity norm before updating model parameters.These operations are presented as the algorithm’s successive steps.
- 3.6. AdaMax: AdaMax does not require initialization-bias correction, and its parameter-update magnitude has a simpler bound than Adam’s.The passage presents both properties as advantages of choosing AdaMax.
3.7. Nadam
Nadam modifies Adam’s momentum component with Nesterov’s accelerated gradient. It is intended to improve convergence speed and the quality of learned models.
- 3.7. Nadam: Nadam combines Adam’s adaptive moment estimation with Nesterov’s accelerated gradient in the momentum component.The supplied passage identifies this modification as Nadam’s defining change.
- 3.7. Nadam: Nadam aims to improve convergence speed and the quality of learned models.These are stated aims rather than reported benchmark results in the supplied passages.
- 3.7. Nadam: Nadam updates first- and second-moment variables, computes corrected moments, and then updates parameters similarly to Adam.The update sequence is referenced through Equations 31 and 32 and subsequent equations.
3.8. AMSGrad
AMSGrad modifies Adam’s normalization by retaining the maximum second-moment estimate, aiming to guarantee convergence while preserving adaptive optimization benefits.
- AMSGrad maintains the maximum of all second-moment estimates up to the current step for normalization.This is its key difference from Adam.
- The method was developed to guarantee convergence while preserving the benefits of Adam and RMSProp.
- Unlike Adam and AdaGrad, AMSGrad neither increases nor decreases the learning rate directly.
4. Experiments
The study compares adaptive optimization algorithms with SGD across supervised and unsupervised image-learning tasks using four datasets and multiple CNN and autoencoder architectures. Adaptive methods generally outperform SGD and AdaGrad, with the strongest results varying by dataset, depth, resolution, and task.
- Experimental setting: Four image datasets—MNIST, CIFAR-10, Kaggle Flowers, and LFW—are evaluated for supervised and unsupervised learning.The supervised experiments use three CNN architectures, while unsupervised experiments use convolutional autoencoders.
- Supervised learning: AdaMax gives the best CIFAR-10 test accuracy across all three CNN architectures.AMSGrad is close for CNN-1 and CNN-2, while Adam and SGD with Nesterov momentum follow AdaMax for deeper CNNs.
- Supervised learning: AdaMax and AMSGrad perform prominently on Kaggle Flowers test accuracy, while SGD gives the worst results.AdaGrad performs comparatively well for CNN-3, especially with high-resolution inputs.
- Supervised learning: Adam, AdaMax, AMSGrad, and AdaDelta generally perform well on test data, and AdaDelta requires no manual learning-rate selection.Momentum improves SGD especially on CIFAR-10, LFW, and Kaggle Flowers.
- Training time: Training-time differences increase with network depth: the gap between the slowest and fastest algorithms reaches nearly 5 minutes for CNN-3.For CNN-1 and CNN-2, the difference is less than 1 minute; AdaMax is the fastest among the algorithms in the cited comparison.
- Unsupervised learning: For unsupervised reconstruction, Adam and its variants generally give the best results, while SGD is by far the worst across both tasks and all datasets.AdaGrad also performs poorly among adaptive algorithms.
- Unsupervised learning: Smaller representations make denoising and color reconstruction harder, while larger representations improve colors reconstructed especially by Adam and its variants.With small representations, none of the algorithms reconstructs Kaggle Flowers colors properly in the cited denoising comparison.
5. Conclusion
The study examines widely used optimization algorithms and compares their behavior and performance across four image datasets, supervised and unsupervised tasks, resolutions, and network architectures.
- The study examines the working principles and pros and cons of widely used deep-learning optimization algorithms.
- Algorithm performance is empirically compared on four image datasets across supervised and unsupervised tasks.
- Training behavior is evaluated under different image resolutions and neural-network architectures.
- Adaptive methods are mostly superior and computationally efficient, with better results for both tasks.
- Research into improved adaptive methods remains ongoing for deep learning.