False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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The unrestricted all-parameter depth separation is false

Statement

For every integer $k \geq 1$ and dimension $d \geq 1$ , the stated Telgarsky bound asserts that there is a function $f (x) = h^{k^{3} + 4} (x_{1})$ , where $h (z) = ρ (2 ρ (z) - 4 ρ (z - 1 / 2))$ , computed by a ReLU network with $2 k^{3} + 8$ layers, $3 k^{3} + 12$ computation nodes, and $4 + d$ distinct parameters, such that for every positive integer triple $t, α, β$ its $L^{1} ([0, 1]^{d})$ distance from every function in the following union is at least $1 / 64$ : functions computed in at most $k$ computational layers by networks with at most $2^{k / (t α β)}$ gate nodes whose gates are $(t, α, β)$ -semi-algebraic; and sums of at most $t$ affine-threshold decision trees, each with at most $2^{k^{3} / t}$ total nodes, including leaves. Here a $(t, α, β)$ -semi-algebraic gate is a map $g : R^{q} \to R$ expressible as a finite sum of polynomial pieces $p_{j}$ of degree at most $β$ , each multiplied by indicators of conjunctions of signs of at most $t$ predicate polynomials $q_{i}$ of degree at most $α$ ; an affine-threshold decision tree recursively branches on tests $⟨ a, x ⟩ - b \geq 0$ and has constant real outputs at its leaves.

Statement references

Telgarsky, Theorem 1.1 and §3.4: depth-separation claim and proof

Telgarsky lecture notes, §5.2: tent-map depth separation

Telgarsky lecture notes, §5.1: Bernstein expectation representation

Telgarsky lecture notes, §5.1: binomial approximation bound used in the refutation

Facts & Assumptions

Given: The claimed bound in the preceding statement.

[F1]

The map $τ (x) = 2 ρ (x) - 4 ρ (x - 1 / 2)$ maps $[0, 1]$ to itself and is $2$ -Lipschitz (Tent-map iterates and ReLU implementation).

[F2]

Every $L$ -Lipschitz function on $[0, 1]$ has degree- $N$ Bernstein polynomial approximation error at most $L / (2 \sqrt{N})$ (Uniform error bound for Bernstein approximation of a Lipschitz function).

Refutation

1.1givenconstructalgebra

Set $d = k = t = α = 1$ , $β = N = 2^{22}$ . The gate $P_{N} (z) = B_{N} (h^{5}) (z)$ is a single $(1, 1, N)$ -semi-algebraic gate: use the constant predicate polynomial $q_{1} (z) = 1$ with condition $q_{1} (z) \geq 0$ , and the one polynomial piece $p_{1} (z) = P_{N} (z)$ , whose degree is at most $N$ . The predicate condition holds for every $z$ , and $q_{1}$ has degree $0 \leq α$ . The claimed node cap is $2^{1 / N} < 2$ , so the integer node count of this network is at most one. Hence $P_{N}$ belongs to the first class in the claimed union.

2.1step 1.1F1F2algebra∎

On $[0, 1]$ , the formula for $h$ reduces to $2 x$ on $[0, 1 / 2]$ and $2 - 2 x$ on $[1 / 2, 1]$ , so it is the tent map in [F1], maps the interval into itself, and is $2$ -Lipschitz. Thus $h^{5}$ is $32$ -Lipschitz. Applying [F2] at $N = 2^{22}$ gives $∥ P_{N} - h^{5} ∥_{\infty} \leq 32 / (2 \sqrt{2^{22}}) = 1 / 128$ . Consequently $\int_{0}^{1} ∣ P_{N} (x) - h^{5} (x) ∣ d x \leq 1 / 128 < 1 / 64$ , contradicting the claimed infimum lower bound.

Depends on

Used by

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Sources

Matus Telgarsky, Benefits of Depth in Neural Networks, Theorem 1.1 and §3.4 (standard reference, not scraped)
Matus Telgarsky, Deep Learning Theory, §5.2 (standard reference, not scraped)