computer science

Proximal gradient methods for learning

Proximal gradient methods for learning is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Proximal gradient methods for learning rather than just read about it. In short: Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies algorithms for a general class of convex regularization problems where the regularization penalty may not be differentiable. One such example is ℓ 1 {\displaystyle \ell _{1}} regularization (also known as Lasso) of the form min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x…

Key takeaways

Proximal gradient methods for learning belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Proximal gradient methods for learning to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Proximal gradient methods for learning from memory before moving on to harder problems.

Reference excerpt

Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies algorithms for a general class of convex regularization problems where the regularization penalty may not be differentiable. One such example is ℓ 1 {\displaystyle \ell _{1}} regularization (also known as Lasso) of the form

min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + λ ‖ w ‖ 1 , where x i ∈ R d and y i ∈ R . {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \|w\|_{1},\quad {\text{ where }}x_{i}\in \mathbb {R} ^{d}{\text{ and }}y_{i}\in \mathbb {R} .}

Proximal gradient methods offer a general framework for solving regularization problems from statistical learning theory with penalties that are tailored to a specific problem application. Such customized penalties can help to induce certain structure in problem solutions, such as sparsity (in the case of lasso) or group structure (in the case of group lasso).

Relevant background Proximal gradient methods are applicable in a wide variety of scenarios for solving convex optimization problems of the form

min x ∈ H F ( x ) + R ( x ) , {\displaystyle \min _{x\in {\mathcal {H}}}F(x)+R(x),}

where F {\displaystyle F} is convex and differentiable with Lipschitz continuous gradient, R {\displaystyle R} is a convex, lower semicontinuous function which is possibly nondifferentiable, and H {\displaystyle {\mathcal {H}}} is some set, typically a Hilbert space. The usual criterion of x {\displaystyle x} minimizes F ( x ) + R ( x ) {\displaystyle F(x)+R(x)} if and only if ∇ ( F + R ) ( x ) = 0 {\displaystyle \nabla (F+R)(x)=0} in the convex, differentiable setting is now replaced by

0 ∈ ∂ ( F + R ) ( x ) , {\displaystyle 0\in \partial (F+R)(x),}

where ∂ φ {\displaystyle \partial \varphi } denotes the subdifferential of a real-valued, convex function φ {\displaystyle \varphi } . Given a convex function φ : H → R {\displaystyle \varphi :{\mathcal {H}}\to \mathbb {R} } an important operator to consider is its proximal operator prox φ : H → H {\displaystyle \operatorname {prox} _{\varphi }:{\mathcal {H}}\to {\mathcal {H}}} defined by

prox φ ⁡ ( u ) = arg ⁡ min x ∈ H φ ( x ) + 1 2 ‖ u − x ‖ 2 2 , {\displaystyle \operatorname {prox} _{\varphi }(u)=\operatorname {arg} \min _{x\in {\mathcal {H}}}\varphi (x)+{\frac {1}{2}}\|u-x\|_{2}^{2},}

which is well-defined because of the strict convexity of the ℓ 2 {\displaystyle \ell _{2}} norm. The proximal operator can be seen as a generalization of a projection. We see that the proximity operator is important because x ∗ {\displaystyle x^{*}} is a minimizer to the problem min x ∈ H F ( x ) + R ( x ) {\displaystyle \min _{x\in {\mathcal {H}}}F(x)+R(x)} if and only if

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Proximal gradient methods for learning

Start with the simplest possible case. Write down what Proximal gradient methods for learning claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Proximal gradient methods for learning before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Proximal gradient methods for learning ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Proximal gradient methods for learning

In research: Proximal gradient methods for learning appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Proximal gradient methods for learning in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Proximal gradient methods for learning is common in secondary-school and first-year university syllabi. It links to neighbouring topics Convex optimization, First order methods, Machine learning, so understanding it makes those chapters shorter.
In everyday life: Look for Proximal gradient methods for learning outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Proximal gradient methods for learning in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Proximal gradient methods for learning means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Proximal gradient methods for learning out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Proximal gradient methods for learning in simple terms?

Why does Proximal gradient methods for learning matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Proximal gradient methods for learning?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Proximal gradient methods for learning.