Learn six methods for handling noisy gradients in stochastic gradient descent algorithms for machine learning and operations research problems.

The technique known as noise injection involves deliberately introducing "noise" into the input data of an ANN during its training phase. A specific form of this technique, called jitter, applies noise to each training instance before each iteration of training. By doing so, the data points "jitter" within the feature space throughout the training, complicating the ANN's ability to precisely fit the original training dataset. This approach helps in mitigating the ANN's tendency to overfit. The effectiveness of this strategy has been validated both theoretically and empirically. The added noise vector typically follows a distribution described by a "kernel."

What are the best methods for handling noisy gradients in stochastic gradient descent algorithms?

Stochastic gradient descent (SGD) is a popular optimization algorithm for machine learning and operations research problems. It iteratively updates the parameters of a model by using a random subset of the data, called a mini-batch, to estimate the gradient of the objective function. However, this approach can introduce noise and variance in the gradient estimates, which can affect the convergence and stability of the algorithm. In this article, you will learn about some of the best methods for handling noisy gradients in SGD algorithms.

1 Noise reduction

One way to reduce the noise in the gradient estimates is to use a larger mini-batch size, which can provide a more accurate approximation of the true gradient. However, this can also increase the computational cost and memory requirements of the algorithm, and may not be feasible for large-scale or distributed problems. Another way to reduce the noise is to use a moving average of the gradients, which can smooth out the fluctuations and outliers. This can be done by using a momentum term, which adds a fraction of the previous gradient update to the current one, or by using an exponential moving average, which weights the past gradients exponentially.

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Ashkan Mohammadi, Ph.D.

Data Scientist | Predictive Modeler | Applied Mathematician | Operation Researcher | Math Olympiad Medalist
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It is almost impossible to have zero noise during a computational procedure. So, the gradient calculation (also known as back-propagation in deep learning) is no exception. A clear example is the finite representation of numbers in computers. For instance, let's say at some point in a calculation, 10 is divided by 3. In the optimization setting, the formulation of a problem can greatly reduce noise during gradient calculations. For example, an objective function with integer coefficients is preferable. However, quite interestingly, sometimes noise in gradients can be beneficial, particularly artificial noises that help escape from saddle points. For instance, consider the noisy gradient descent algorithms.

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2 Noise adaptation

Another way to handle noisy gradients is to adapt the learning rate, which controls how much the parameters are updated in each iteration, according to the noise level. This can help to avoid overshooting or undershooting the optimal solution, and to balance the trade-off between exploration and exploitation. Some examples of noise-adaptive learning rate methods are AdaGrad, which scales the learning rate inversely proportional to the accumulated squared gradients, RMSProp, which scales the learning rate inversely proportional to the moving average of the squared gradients, and Adam, which combines both ideas and also uses a bias correction term.

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3 Noise injection

A third way to handle noisy gradients is to inject artificial noise into the gradient updates, which can have a beneficial effect on the algorithm's performance. This can help to escape from local minima or saddle points, to regularize the model and prevent overfitting, and to improve the generalization and robustness of the model. Some examples of noise injection methods are stochastic gradient Langevin dynamics, which adds Gaussian noise proportional to the inverse square root of the learning rate, dropout, which randomly drops out some units or connections in the model, and noisy activation functions, which add noise to the outputs of the hidden layers.

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Bertrand K. Hassani, PhD

GROUP CEO - Global Leader in Analytics, AI, Data Science, Research, and Innovation
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The technique known as noise injection involves deliberately introducing "noise" into the input data of an ANN during its training phase. A specific form of this technique, called jitter, applies noise to each training instance before each iteration of training. By doing so, the data points "jitter" within the feature space throughout the training, complicating the ANN's ability to precisely fit the original training dataset. This approach helps in mitigating the ANN's tendency to overfit. The effectiveness of this strategy has been validated both theoretically and empirically. The added noise vector typically follows a distribution described by a "kernel."

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4 Noise analysis

A fourth way to handle noisy gradients is to analyze the noise characteristics and sources, and to design or choose an algorithm that is suitable for the problem at hand. This can help to understand the behavior and performance of the algorithm, and to tune the hyperparameters accordingly. Some factors that can influence the noise level are the data distribution, the model complexity, the objective function, and the optimization algorithm. Some tools that can help to analyze the noise are gradient variance estimators, which can measure the variance of the gradient estimates, and gradient noise scale, which can measure the ratio of the gradient noise to the gradient signal.

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5 Noise mitigation

A fifth way to handle noisy gradients is to mitigate the negative effects of the noise on the algorithm's convergence and stability. This can help to improve the efficiency and reliability of the algorithm, and to avoid divergence or oscillation. Some examples of noise mitigation methods are gradient clipping, which limits the norm of the gradient update to a threshold, gradient normalization, which divides the gradient update by its norm, and gradient projection, which projects the gradient update onto a feasible set.

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6 Noise exploitation

A sixth way to handle noisy gradients is to exploit the positive effects of the noise on the algorithm's exploration and diversity. This can help to find better or multiple solutions, and to avoid premature convergence or stagnation. Some examples of noise exploitation methods are simulated annealing, which gradually decreases the noise level as the algorithm progresses, evolutionary algorithms, which use noise to generate and select new solutions, and ensemble methods, which use noise to create and combine different models.

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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What are the best methods for handling noisy gradients in stochastic gradient descent algorithms?

1

2

3

4

5

6

7

1 Noise reduction

2 Noise adaptation

3 Noise injection

4 Noise analysis

5 Noise mitigation

6 Noise exploitation

7 Here’s what else to consider

Operations Research

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What are the best methods for handling noisy gradients in stochastic gradient descent algorithms?

1

2

3

4

5

6

7

1 Noise reduction

2 Noise adaptation

3 Noise injection

4 Noise analysis

5 Noise mitigation

6 Noise exploitation

7 Here’s what else to consider

Operations Research

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