Gradient descent

23 languages

• Article
• Talk

• Read
• Edit
• View history

From Wikipedia, the free encyclopedia

For the analytical method called "steepest descent", see Method of steepest descent.

Part of a series on
Machine learning and data mining
show Paradigms
show Problems
show Supervised learning (classification • regression)
show Clustering
show Dimensionality reduction
show Structured prediction
show Anomaly detection
show Artificial neural network
show Reinforcement learning
show Learning with humans
show Model diagnostics
show Theory
show Machine-learning venues
show Related articles
v t e

0:14

Gradient Descent in 2D

In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent.

Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847.^[1] Jacques Hadamard independently proposed a similar method in 1907.^[2][3] Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944,^[4] with the method becoming increasingly well-studied and used in the following decades.^[5][6]

Description[edit]

Illustration of gradient descent on a series of level sets

Gradient descent is based on the observation that if the multi-variable function ${\displaystyle �(\mathbf{�})}$ is defined and differentiable in a neighborhood of a point ${\displaystyle \mathbf{�}}$, then ${\displaystyle �(\mathbf{�})}$ decreases fastest if one goes from ${\displaystyle \mathbf{�}}$ in the direction of the negative gradient of ${\displaystyle �}$ at ${\displaystyle \mathbf{�},-\mathrm{\nabla }�(\mathbf{�})}$. It follows that, if

${\displaystyle {\mathbf{�}}_{�+1}={\mathbf{�}}_{�}-�\mathrm{\nabla }�({\mathbf{�}}_{�})}$

for a small enough step size or learning rate ${\displaystyle �\in {\mathbb{�}}_{+}}$, then ${\displaystyle �({\mathbf{�}}_{\mathbf{�}})\ge �({\mathbf{�}}_{\mathbf{�}\mathbf{+}\mathbf{1}})}$. In other words, the term ${\displaystyle �\mathrm{\nabla }�(\mathbf{�})}$ is subtracted from ${\displaystyle \mathbf{�}}$ because we want to move against the gradient, toward the local minimum. With this observation in mind, one starts with a guess ${\displaystyle {\mathbf{�}}_0}$ for a local minimum of ${\displaystyle �}$, and considers the sequence ${\displaystyle {\mathbf{�}}_0,{\mathbf{�}}_1,{\mathbf{�}}_2,\dots }$ such that

${\displaystyle {\mathbf{�}}_{�+1}={\mathbf{�}}_{�}-{�}_{�}\mathrm{\nabla }�({\mathbf{�}}_{�}),�\ge 0.}$

We have a monotonic sequence

${\displaystyle �({\mathbf{�}}_0)\ge �({\mathbf{�}}_1)\ge �({\mathbf{�}}_2)\ge \cdots ,}$

so, hopefully, the sequence ${\displaystyle ({\mathbf{�}}_{�})}$ converges to the desired local minimum. Note that the value of the step size ${\displaystyle �}$ is allowed to change at every iteration. With certain assumptions on the function ${\displaystyle �}$ (for example, ${\displaystyle �}$ convex and ${\displaystyle \mathrm{\nabla }�}$ Lipschitz) and particular choices of ${\displaystyle �}$ (e.g., chosen either via a line search that satisfies the Wolfe conditions, or the Barzilai–Borwein method^[7][8] shown as following),

${\displaystyle {�}_{�}=\frac{\left|{\left({\mathbf{�}}_{�}-{\mathbf{�}}_{�-1}\right)}^{�}\left[\mathrm{\nabla }�({\mathbf{�}}_{�})-\mathrm{\nabla }�({\mathbf{�}}_{�-1})\right]\right|}{{\Vert \mathrm{\nabla }�({\mathbf{�}}_{�})-\mathrm{\nabla }�({\mathbf{�}}_{�-1})\Vert }^2}}$

convergence to a local minimum can be guaranteed. When the function ${\displaystyle �}$ is convex, all local minima are also global minima, so in this case gradient descent can converge to the global solution.

This process is illustrated in the adjacent picture. Here, ${\displaystyle �}$ is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines, that is, the regions on which the value of ${\displaystyle �}$ is constant. A red arrow originating at a point shows the direction of the negative gradient at that point. Note that the (negative) gradient at a point is orthogonal to the contour line going through that point. We see that gradient descent leads us to the bottom of the bowl, that is, to the point where the value of the function ${\displaystyle �}$ is minimal.

An analogy for understanding gradient descent[edit]

Fog in the mountains

The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. A person is stuck in the mountains and is trying to get down (i.e., trying to find the global minimum). There is heavy fog such that visibility is extremely low. Therefore, the path down the mountain is not visible, so they must use local information to find the minimum. They can use the method of gradient descent, which involves looking at the steepness of the hill at their current position, then proceeding in the direction with the steepest descent (i.e., downhill). If they were trying to find the top of the mountain (i.e., the maximum), then they would proceed in the direction of steepest ascent (i.e., uphill). Using this method, they would eventually find their way down the mountain or possibly get stuck in some hole (i.e., local minimum or saddle point), like a mountain lake. However, assume also that the steepness of the hill is not immediately obvious with simple observation, but rather it requires a sophisticated instrument to measure, which the person happens to have at the moment. It takes quite some time to measure the steepness of the hill with the instrument, thus they should minimize their use of the instrument if they wanted to get down the mountain before sunset. The difficulty then is choosing the frequency at which they should measure the steepness of the hill so not to go off track.

In this analogy, the person represents the algorithm, and the path taken down the mountain represents the sequence of parameter settings that the algorithm will explore. The steepness of the hill represents the slope of the function at that point. The instrument used to measure steepness is differentiation. The direction they choose to travel in aligns with the gradient of the function at that point. The amount of time they travel before taking another measurement is the step size.

Choosing the step size and descent direction[edit]

Since using a step size ${\displaystyle �}$ that is too small would slow convergence, and a ${\displaystyle �}$ too large would lead to divergence, finding a good setting of ${\displaystyle �}$ is an important practical problem. Philip Wolfe also advocated using "clever choices of the [descent] direction" in practice.^[9] Whilst using a direction that deviates from the steepest descent direction may seem counter-intuitive, the idea is that the smaller slope may be compensated for by being sustained over a much longer distance.

To reason about this mathematically, consider a direction ${\displaystyle {\mathbf{�}}_{�}}$ and step size ${\displaystyle {�}_{�}}$ and consider the more general update:

${\displaystyle {\mathbf{�}}_{�+1}={\mathbf{�}}_{�}-{�}_{�}{\mathbf{�}}_{�}}$.

Finding good settings of ${\displaystyle {\mathbf{�}}_{�}}$ and ${\displaystyle {�}_{�}}$ requires some thought. First of all, we would like the update direction to point downhill. Mathematically, letting ${\displaystyle {�}_{�}}$ denote the angle between ${\displaystyle -\mathrm{\nabla }�({\mathbf{�}}_{\mathbf{�}})}$ and ${\displaystyle {\mathbf{�}}_{�}}$, this requires that  To say more, we need more information about the objective function that we are optimising. Under the fairly weak assumption that ${\displaystyle �}$ is continuously differentiable, we may prove that:^[10]

${\displaystyle �({\mathbf{�}}_{�+1})\le �({\mathbf{�}}_{�})-{�}_{�}\Vert \mathrm{\nabla }�({\mathbf{�}}_{�}){\Vert }_2\Vert {\mathbf{�}}_{�}{\Vert }_2\left[\cos {�}_{�}-\underset{�\in [0,1]}{max}\frac{\Vert \mathrm{\nabla }�({\mathbf{�}}_{�}-�{�}_{�}{\mathbf{�}}_{�})-\mathrm{\nabla }�({\mathbf{�}}_{�}){\Vert }_2}{\Vert \mathrm{\nabla }�({\mathbf{�}}_{�}){\Vert }_2}\right]}$

(1)

This inequality implies that the amount by which we can be sure the function ${\displaystyle �}$ is decreased depends on a trade off between the two terms in square brackets. The first term in square brackets measures the angle between the descent direction and the negative gradient. The second term measures how quickly the gradient changes along the descent direction.

In principle inequality (1) could be optimized over ${\displaystyle {\mathbf{�}}_{�}}$ and ${\displaystyle {�}_{�}}$ to choose an optimal step size and direction. The problem is that evaluating the second term in square brackets requires evaluating ${\displaystyle \mathrm{\nabla }�({\mathbf{�}}_{�}-�{�}_{�}{\mathbf{�}}_{�})}$, and extra gradient evaluations are generally expensive and undesirable. Some ways around this problem are:

• Forgo the benefits of a clever descent direction by setting ${\displaystyle {\mathbf{�}}_{�}=\mathrm{\nabla }�({\mathbf{�}}_{\mathbf{�}})}$, and use line search to find a suitable step-size ${\displaystyle {�}_{�}}$, such as one that satisfies the Wolfe conditions. A more economic way of choosing learning rates is backtracking line search, a method that has both good theoretical guarantees and experimental results. Note that one does not need to choose ${\displaystyle {\mathbf{�}}_{�}}$ to be the gradient; any direction that has positive intersection product with the gradient will result in a reduction of the function value (for a sufficiently small value of ${\displaystyle {�}_{�}}$).
• Assuming that ${\displaystyle �}$ is twice-differentiable, use its Hessian ${\displaystyle {\mathrm{\nabla }}^2�}$ to estimate ${\displaystyle \Vert \mathrm{\nabla }�({\mathbf{�}}_{�}-�{�}_{�}{\mathbf{�}}_{�})-\mathrm{\nabla }�({\mathbf{�}}_{�}){\Vert }_2\approx \Vert �{�}_{�}{\mathrm{\nabla }}^2�({\mathbf{�}}_{�}){\mathbf{�}}_{�}\Vert .}$Then choose ${\displaystyle {\mathbf{�}}_{�}}$ and ${\displaystyle {�}_{�}}$ by optimising inequality (1).
• Assuming that ${\displaystyle \mathrm{\nabla }�}$ is Lipschitz, use its Lipschitz constant ${\displaystyle �}$ to bound ${\displaystyle \Vert \mathrm{\nabla }�({\mathbf{�}}_{�}-�{�}_{�}{\mathbf{�}}_{�})-\mathrm{\nabla }�({\mathbf{�}}_{�}){\Vert }_2\le ��{�}_{�}\Vert {\mathbf{�}}_{�}\Vert .}$ Then choose ${\displaystyle {\mathbf{�}}_{�}}$ and ${\displaystyle {�}_{�}}$ by optimising inequality (1).
• Build a custom model of ${\displaystyle \underset{�\in [0,1]}{max}\frac{\Vert \mathrm{\nabla }�({\mathbf{�}}_{�}-�{�}_{�}{\mathbf{�}}_{�})-\mathrm{\nabla }�({\mathbf{�}}_{�}){\Vert }_2}{\Vert \mathrm{\nabla }�({\mathbf{�}}_{�}){\Vert }_2}}$ for ${\displaystyle �}$. Then choose ${\displaystyle {\mathbf{�}}_{�}}$ and ${\displaystyle {�}_{�}}$ by optimising inequality (1).
• Under stronger assumptions on the function ${\displaystyle �}$ such as convexity, more advanced techniques may be possible.

Usually by following one of the recipes above, convergence to a local minimum can be guaranteed. When the function ${\displaystyle �}$ is convex, all local minima are also global minima, so in this case gradient descent can converge to the global solution.