Author: Akash Ajagekar (SYSEN 6800 Fall 2021)

## Background

### Batch Gradient Descent

In standard batch gradient descent, the parameters, ${\displaystyle \theta }$, of the objective function ${\displaystyle f(\theta )}$, are updated based on the gradient of ${\displaystyle f}$ with respect to ${\displaystyle \theta }$ for the entire training dataset, as

${\displaystyle g_{t}=\nabla _{\theta _{t-1}}f{\big (}\theta _{t-1}{\big )}}$
${\displaystyle \theta _{t}=\theta _{t-1}-\alpha g_{t},}$

where ${\displaystyle \alpha }$ is defined as the learning rate and is a hyper-parameter of the optimization algorithm, and ${\displaystyle t}$ is the iteration number. Key challenges of the standard gradient descent method are the tendency to get stuck in local minima and/or saddle points of the objective function, as well as choosing a proper learning rate, ${\displaystyle \alpha }$, which can lead to poor convergence.[2]

### Stochastic Gradient Descent

Another variant of gradient descent is stochastic gradient descent (SGD), the gradient is computed and parameters are updated as in equation 1, but for each training sample in the training set.

### Mini-Batch Gradient Descent

In between batch gradient descent and stochastic gradient descent, mini-batch gradient descent computes parameters updates on the gradient computed from a subset of the training set, where the size of the subset is often referred to as the batch size.

The Adam algorithm first computes the gradient, ${\displaystyle g_{t}}$ of the objective function with respect to the parameters ${\displaystyle \theta }$, but then computes and stores first and second order moments of the gradient, ${\displaystyle m_{t}}$ and ${\displaystyle v_{t}}$ respectively, as

${\displaystyle m_{t}=\beta _{1}\cdot m_{t-1}+(1-\beta _{1})\cdot g_{t}}$
${\displaystyle v_{t}=\beta _{2}\cdot v_{t-1}+(1-\beta _{2})\cdot g_{t}^{2},}$

where ${\displaystyle \beta _{1}}$ and ${\displaystyle \beta _{2}}$ are hyper-parameters that are ${\displaystyle \in [0,1]}$. These parameters can seen as exponential decay rates of the estimated moments, as the previous value is successively multiplied by the value less than 1 in each iteration. The authors of the original paper suggest values ${\displaystyle \beta _{1}=0.9}$ and ${\displaystyle \beta _{2}=0.999}$. In the current notation, the first iteration of the algorithm is at ${\displaystyle t=1}$ and both, ${\displaystyle m_{0}}$ and ${\displaystyle v_{0}}$ are initialized to zero. Since both moments are initialized to zero, at early time steps, these values are biased towards zero. To counter this, the authors proposed a corrected update to ${\displaystyle m_{t}}$ and ${\displaystyle v_{t}}$ as

${\displaystyle {\hat {m}}_{t}=m_{t}/(1-\beta _{1}^{t})}$
${\displaystyle {\hat {v}}_{t}=v_{t}/(1-\beta _{2}^{t}).}$
Finally, the parameter update is computed as

${\displaystyle \theta _{t}=\theta _{t-1}-\alpha \cdot {\hat {m}}_{t}/({\sqrt {{\hat {v}}_{t}}}+\epsilon ),}$

where ${\displaystyle \epsilon }$ is a small constant for stability. The authors recommend a value of ${\displaystyle \epsilon =10^{-8}}$.

## Numerical Example

To illustrate how updates occur in the Adam algorithm, consider a linear, least-squares regression problem formulation. The table below shows a sample data-set of student exam grades and the number of hours spent studying for the exam. The goal of this example will be to generate a linear model to predict exam grades as a function of time spent studying.

 Hours Studying 9 4.9 1.6 1.9 7.9 2 11.5 3.9 1.1 1.6 5.1 8.2 7.3 10.4 11.2 Exam Grad 88 72.3 66.5 65.1 79.5 60.8 94.3, 66.7 65.4 63.8 68.4 82.5 75.9 87.8 85.2

The hypothesized model function will be

${\displaystyle f_{\theta }(x)=\theta _{0}+\theta _{1}x.}$

The cost function is defined as

${\displaystyle J({\theta })={\frac {1}{2}}\sum _{i}^{n}{\big (}f_{\theta }(x_{i})-y_{i}{\big )}^{2},}$

Where the ${\displaystyle 1/2}$ coefficient is used only to make the derivatives cleaner. The optimization problem can then be formulated as trying to find the values of ${\displaystyle \theta }$ that minimize the squared residuals of ${\displaystyle f_{\theta }(x)}$ and ${\displaystyle y}$.

${\displaystyle \mathrm {argmin} _{\theta }\quad {\frac {1}{n}}\sum _{i}^{n}{\big (}f_{\theta }(x_{i})-y_{i}{\big )}^{2}}$

For simplicity, parameters will be updated after every data point i.e. a batch size of 1. For a single data point the derivatives of the cost function with respect to ${\displaystyle \theta _{0}}$ and ${\displaystyle \theta _{1}}$ are

${\displaystyle {\frac {\partial J(\theta )}{\partial \theta _{0}}}={\big (}f_{\theta }(x)-y{\big )}}$
${\displaystyle {\frac {\partial J(\theta )}{\partial \theta _{1}}}={\big (}f_{\theta }(x)-y{\big )}x}$

The initial values of ${\displaystyle {\theta }}$ will be set to [50, 1] and The learning rate, ${\displaystyle \alpha }$, is set to 0.1 and the suggested parameters for ${\displaystyle \beta _{1}}$, ${\displaystyle \beta _{2}}$, and ${\displaystyle \epsilon }$ are used. With the first data sample of ${\displaystyle (x,y)=[8.98,88.01]}$, the computed gradients are

${\displaystyle {\frac {\partial J(\theta )}{\partial \theta _{0}}}={\big (}(50+1\cdot 9-88.01{\big )}=-29.0}$
${\displaystyle {\frac {\partial J(\theta )}{\partial \theta _{1}}}={\big (}(50+1\cdot 9-88.01{\big )}\cdot 9.0=-261}$

With ${\displaystyle m_{0}}$ and ${\displaystyle v_{0}}$ being initialized to zero, the calculations of ${\displaystyle m_{1}}$ and ${\displaystyle v_{1}}$ are

${\displaystyle m_{1}=0.9\cdot 0+(1-0.9)\cdot {\begin{bmatrix}-29\\-261\end{bmatrix}}={\begin{bmatrix}-2.9\\-26.1\end{bmatrix}}}$
${\displaystyle v_{1}=0.999\cdot 0+(1-0.999)\cdot {\begin{bmatrix}-29^{2}\\-261^{2}\end{bmatrix}}={\begin{bmatrix}0.84\\68.2\end{bmatrix}},}$

The bias-corrected terms are computed as

${\displaystyle {\hat {m}}_{1}={\begin{bmatrix}-2.9\\-26.1\end{bmatrix}}{\frac {1}{(1-0.9^{1})}}={\begin{bmatrix}-29.0\\-261.1\end{bmatrix}}}$
${\displaystyle {\hat {v}}_{1}={\begin{bmatrix}0.84\\68.2\end{bmatrix}}{\frac {1}{(1-0.999^{1})}}={\begin{bmatrix}851.5\\68168\end{bmatrix}}.}$

Finally, the parameter update is

${\displaystyle \theta _{0}=50-0.1\cdot -29/({\sqrt {851.5}}+10^{-8})=50.1}$
${\displaystyle \theta _{1}=1-0.1\cdot -261/({\sqrt {68168}}+10^{-8})=1.1}$

This procedure is repeated until the parameters have converged, giving ${\displaystyle \theta }$ values of ${\displaystyle [58.98,2.72]}$. The figures to the right show the trajectory of the Adam algorithm over a contour plot of the objective function and the resulting model fit. It should be noted that the stochastic gradient descent algorithm with a learning rate of 0.1 diverges and with a rate of 0.01, SGD oscillates around the global minimum due to the large magnitudes of the gradient in the ${\displaystyle \theta _{1}}$ direction.

## Applications

The Adam optimization algorithm has been widely used in machine learning applications to train model parameters. When used with backpropagation, the Adam algorithm has been shown to be a very robust and efficient method for training artificial neural networks and is capable of working well with a variety of structures and applications. In their original paper, the authors present three different training examples, logistic regression, multi-layer neural networks for classification of MNIST images, and a convolutional neural network (CNN). The training results from the original Adam paper showing the objective function cost vs. the iteration over the entire data set for the multi-layer neural network is shown to the right.

## Variants of Adam

AdaMax[3] is a variant of the Adam algorithm proposed in the original Adam paper that uses an exponentially weighted infinity norm instead of the second-order moment estimate. The weighted infinity norm updated ${\displaystyle u_{t}}$, is computed as

${\displaystyle u_{t}=\max(\beta _{2}\cdot u_{t-1},|g_{t}|).}$

The parameter update then becomes

${\displaystyle \theta _{t}=\theta _{t-1}-(\alpha /(1-\beta _{1}^{t}))\cdot m_{t}/u_{t}.}$

The Nadam algorithm[4] was proposed in 2016 and incorporates the Nesterov Accelerate Gradient (NAG)[5], a popular momentum like SGD variation, into the first-order moment term.

## Conclusion

Adam is a variant of the gradient descent algorithm that has been widely adopted in the machine learning community. Adam can be seen as the combination of two other variants of gradient descent, SGD with momentum and RMSProp. Adam uses estimations of the first and second-order moments of the gradient to adapt the parameter update. These moment estimations are computed via moving averages,${\displaystyle m_{t}}$ and ${\displaystyle v_{t}}$, of the gradient and the squared gradient respectfully. In a variety of neural network training applications, Adam has shown increased convergence and robustness over other gradient descent algorithms and is often recommended as the default optimizer for training.[6]

## References

1. https://arxiv.org/pdf/1412.6980.pdf ADAM: A METHOD FOR STOCHASTIC OPTIMIZATION
2. Ruder, Sebastian. An Overview of Gradient Descent Optimization Algorithms, 2016, pp. 1–14, http://arxiv.org/abs/1609.04747.
3. Cite error: Invalid <ref> tag; no text was provided for refs named adam
4. Dozat, Timothy. Incorporating Nesterov Momentum into Adam. ICLR Workshop, no. 1, 2016, pp. 2013–16.
5. Nesterov, Yuri. A method of solving a convex programming problem with convergence rate O(1/k^2). In Soviet Mathematics Doklady, 1983, pp. 372-376.
6. "Neural Networks Part 3: Learning and Evaluation," CS231n: Convolutional Neural Networks for Visual Recognition, Stanford Unversity, 2020