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Gram control and a Taylor remainder imply one-step loss contraction

Statement

Let $r \in R^{n}$ , let $G$ be a real symmetric matrix with eigenvalues in $[λ, Λ]$ for some $0 < λ \leq Λ$ , and suppose $0 < η Λ \leq 1$ . If an update has the form $r^{+} = (I - η G) r + R$ with $∥ R ∥_{2} \leq η λ ∥ r ∥_{2} / 4$ , then for the quadratic loss $L = \frac{1}{2} ∥ r ∥_{2}^{2}$ and $L^{+} = \frac{1}{2} ∥ r^{+} ∥_{2}^{2}$ ,

$L^{+} \leq (1 - η λ / 2) L .$

Statement references

Du et al., empirical loss contraction, Theorems 5.1 and 6.1

Facts & Assumptions

Given: The residual vector, symmetric Gram matrix, eigenvalue bounds, step size, and remainder bound.

[F1]

A real symmetric matrix has an orthonormal eigenbasis. Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis

[F2]

The Euclidean norm satisfies the triangle inequality. Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation

Proof

technique · bound the spectral norm of the linear contraction and then square the residual-norm bound

1.1F1F2givenalgebra

By [F1], the eigenvalues $μ_{j}$ of $G$ lie in $[λ, Λ]$ , so those of $I - η G$ lie in $[1 - η Λ, 1 - η λ] \subseteq [0, 1 - η λ]$ . Expanding $r$ in an orthonormal eigenbasis gives $∥ (I - η G) r ∥_{2}^{2} = \sum_{j} (1 - η μ_{j})^{2} r_{j}^{2} \leq (1 - η λ)^{2} ∥ r ∥_{2}^{2}$ . The norm triangle inequality [F2] with the remainder hypothesis then yields $∥ r^{+} ∥_{2} \leq (1 - 3 η λ / 4) ∥ r ∥_{2}$ .

2.1step 1.1algebra∎

Put $t = η λ \in (0, 1]$ . Then $(1 - 3 t / 4)^{2} \leq 1 - t / 2$ , since $1 - t / 2 - (1 - 3 t / 4)^{2} = t (1 - 9 t / 16) \geq 7 t / 16 > 0$ . Squaring the estimate in step 1.1 and multiplying by $1 / 2$ yields $L^{+} \leq (1 - η λ / 2) L$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Simon S. Du, Jason D. Lee, Haochuan Li, Liwei Wang, and Xiyu Zhai, Gradient Descent Finds Global Minima of Deep Neural Networks, §§4–6 (standard reference, not scraped)