Suppose
𝑔
(
𝑤
|
𝑦
𝑖
,
𝑥
𝑖
)
=
(
𝑦
𝑖
−
𝑥
𝑖
𝑇
𝑤
)
2
, where
𝑤
∈
𝑅
𝑝
,
𝑦
𝑖
∈
𝑅
, and
𝑥
𝑖
∈
𝑅
𝑝
, for
𝑖
=
1
…
𝑛
. Is
∑
𝑖
=
1
𝑛
𝑔
(
𝑤
|
𝑦
𝑖
,
𝑥
𝑖
)
convex?

Question

Suppose 
𝑔
(
𝑤
|
𝑦
𝑖
,
𝑥
𝑖
)
=
(
𝑦
𝑖
−
𝑥
𝑖
𝑇
𝑤
)
2
, where 
𝑤
∈
𝑅
𝑝
, 
𝑦
𝑖
∈
𝑅
, and 
𝑥
𝑖
∈
𝑅
𝑝
, for 
𝑖
=
1
…
𝑛
. Is 
∑
𝑖
=
1
𝑛
𝑔
(
𝑤
|
𝑦
𝑖
,
𝑥
𝑖
)
 convex?

BlackTom AI · Accepted Answer

To evaluate convexity, I’ll look at the function in terms of w with the given data fixed.
Option 1: 'No' — This would claim the sum is not convex. However, each term g(w|y_i,x_i) = (y_i - x_i^T w)^2 is a quadratic function in w with Hessian 2 x_i x_i^T, which is positive semidefinite for any x_i. The sum of convex (indeed quadratic) functions is convex, so this option is incorrect.
Option 2: 'Not enough information is provided to answer the question.' Here, we know all x_i and y_i (for the functional form), and the convexity argument relies only on the fact that each term is a convex quadratic in w. There is no dependence on additional unspecified parameters that could change convexity, so we do have enough information to answer. This statement is therefore incorrect.
Option 3: 'Yes.' Each term is a convex function of w because its Hessian 2 x_i x_i^T is PSD, and the sum of PSD (or more generally convex) functions is convex. Even if some x_i are zero, the Hessian remains PSD (possibly zero), which still preserves convexity. Therefore the sum ∑ g(w|y_i,x_i) is convex in w, making this option correct.
In summary, the only statement that aligns with the mathematical property of convexity for a sum of squared linear forms is Option 3: Yes. The alternative statements fail by misjudging convexity, either by claiming non-convexity or by asserting insufficient information without recognizing the inherent convexity of each quadratic term.

25S-STATS-102B-LEC-3 S25 Midterm Exam- Requires Respondus LockDown Browser

Suppose 𝑔 ( 𝑤 | 𝑦 𝑖 , 𝑥 𝑖 ) = ( 𝑦 𝑖 − 𝑥 𝑖 𝑇 𝑤 ) 2 , where 𝑤 ∈ 𝑅 𝑝 , 𝑦 𝑖 ∈ 𝑅 , and 𝑥 𝑖 ∈ 𝑅 𝑝 , for 𝑖 = 1 … 𝑛 . Is ∑ 𝑖 = 1 𝑛 𝑔 ( 𝑤 | 𝑦 𝑖 , 𝑥 𝑖 ) convex?

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