This math sits underneath every AI model being trained right now.
Gradient. Jacobian. Hessian.
Three words that look intimidating at first.
But they are really just three ways of measuring change.
𝟭. 𝗚𝗿𝗮𝗱𝗶𝗲𝗻𝘁 ∇f
Takes a scalar function:
f : ℝⁿ → ℝ
Returns a vector of first-order partial derivatives.
It answers:
"Which direction makes f increase fastest?"
That is why gradients are central to optimization.
Gradient descent moves in the opposite direction because the gradient points uphill.
Backpropagation efficiently computes gradients during training.
𝟮. 𝗝𝗮𝗰𝗼𝗯𝗶𝗮𝗻 J_F
Takes a vector-valued function:
F : ℝⁿ → ℝᵐ
Returns an m × n matrix of first-order partial derivatives.
It answers:
"How does each output change with each input?"
The Jacobian is the local linear map of a vector-valued function.
It shows up in:
→ sensitivity analysis
→ change of variables
→ automatic differentiation
→ forward-mode AD
→ reverse-mode AD / backpropagation
In simple terms:
forward-mode AD uses Jacobian-vector products.
reverse-mode AD uses vector-Jacobian products.
𝟯. 𝗛𝗲𝘀𝘀𝗶𝗮𝗻 H_f
Takes a scalar function:
f : ℝⁿ → ℝ
Returns an n × n matrix of second-order partial derivatives.
It answers:
"How does the gradient itself change?"
That means the Hessian measures curvature.
When the second partial derivatives are continuous, the Hessian is symmetric.
At a critical point:
→ positive definite Hessian → strict local minimum
→ negative definite Hessian → strict local maximum
→ indefinite Hessian → saddle point
The clean mental model
Gradient = first derivatives of one output
→ tells you direction
Jacobian = first derivatives of many outputs
→ tells you sensitivity
Hessian = second derivatives of one output
→ tells you curvature
And the relationship between them is simple:
The Hessian is the Jacobian of the gradient.
For a scalar output, the Jacobian contains the same partial derivatives as the gradient, up to row/column convention.
Same idea:
measure change.
Different object:
direction, sensitivity, curvature.
Once this clicks, optimization stops looking like a pile of formulas.
It starts looking like a map of the problem.
Where could we improve Composer 2.5?
We're working on the next model and would love your feedback.
Lots of work to do (our CursorBench evals below) in the coming weeks!
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