## How do you derive subgradient?

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- solution: to find a subgradient at x, • if f(x)=0 (that is, x ∈ C), take g = 0.
- • if f(x) > 0, find projection y = P(x) on C; take. g =
- �y− x�2(x − y) =
- �x − P(x)�2(x − P(x))

**Is subgradient convex?**

The subgradient The subdifferential is always a nonempty convex compact set. The set of all subgradients at x0 is called the subdifferential at x0 and is again denoted ∂f(x0).

### How do you find the normal cone?

Normal cone: given any set C and point x ∈ C, we can define normal cone as NC(x) = {g : gT x ≥ gT y for all y ∈ C} Normal cone is always a convex cone. Positive semidefinite: a matrix X is positive semidefinite if all the eigenvalues of X are larger or equal to 0 ⇐⇒ aT Xa ≥ 0 for all a ∈ Rn.

**What is the conjugate of a function?**

1. The conjugate function f* (y) is the maximum gap between the linear function yx and the function f (x). Example 1 (Affine function) f (x) = ax + b. By definition, the conjugate function is given by f∗ (y) = supx (yx − ax − b).

## What is strongly convex?

Intuitively speaking, strong convexity means that there exists a quadratic lower bound on the growth of the function. This directly implies that a strong convex function is strictly convex since the quadratic lower bound growth is of course strictly grater than the linear growth.

**What is Subdifferential of a convex function?**

The subgradient The set of all subgradients at x0 is called the subdifferential at x0 and is denoted ∂f(x0). The subdifferential is always a nonempty convex compact set. These concepts generalize further to convex functions f:U→ R on a convex set in a locally convex space V.

### What is the polar of a cone?

It can be seen that the polar cone is equal to the negative of the dual cone, i.e. Co = −C*. For a closed convex cone C in X, the polar cone is equivalent to the polar set for C.

**Is cone a convex set?**

In this context, a convex cone is a cone that is closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets.

## What is a λ strongly convex function?

The strong convexity parameter λ is a measure of the curvature of f. By rearranging terms, this tells us that a λ-strong convex function can be lower bounded by the following inequality: f(x) ≥ f(y) − ∇f(y)T (y − x) +

**What is beta smoothness?**

Smoothness. Definition A continuously differentiable function f is β-smooth if the gradient ∇f is β-Lipschitz, that is if for all x, y ∈ X, ∇f (y) − ∇f (x) ≤ βy − x . Property If f is β-smooth, then for any x, y ∈ X: ∣ ∣f (y) − f (x) − 〈∇f (x), y − x〉 ∣ ∣ ≤ β 2 y − x2 .

### What is a polyhedral set?

A polyhedron is the set of solution points of a linear system: Sol(A · x ≤ b) = {x0 ∈ R|x| | A · x0 ≤ b}. Polyhedra are convex sets. The homogeneous version of a linear system H(A·x ≤ b) = A·x ≤ 0 is the linear system where constant terms are replaced by 0’s.

**Does L2 normalization have anything to do with L2 regularization?**

Does L2 normalization have anything to do with L2 regularization? L2 regularization operates on the parameters of a model, whereas L2 normalization (in the context you’re asking about) operates on the representation of the data.

## What is L2 normalization in tf-idf?

Both classes [TfidfTransformer and TfidfVectorizer] also apply L2 normalization after computing the tf-idf representation; in other words, they rescale the representation of each document to have Euclidean norm 1. Rescaling in this way means that the length of a document (the number of words) does not change the vectorized representation.

**What is normalization in machine learning?**

Improving Performance of ML Model (Contd…) It may be defined as the normalization technique that modifies the dataset values in a way that in each row the sum of the squares will always be up to 1. It is also called least squares.

### What is a sub gradient of F at x?

Geometrically, g is a subgradient of f at x if (g,−1) supports epif at (x,f(x)), as illustrated in ﬁgure 2. A function f is called subdiﬀerentiable at x if there exists at least one subgradient at x. The set of subgradients of f at the point x is called the subdiﬀerential of f at x, and is denoted ∂f(x).