Newman's Theorem in Communication Complexity

• Newman's theorem is a key result in communication complexity, demonstrating that public randomness can be simulated by private randomness with a logarithmic additional communication cost.
• The proof utilizes classical and approximate Carathéodory theorems to efficiently compress and sparsify the set of deterministic protocols while maintaining control over the L∞-error.
• This geometric approach replaces traditional probabilistic methods, emphasizing that public coins offer only a limited advantage in randomized protocol design.

Searching arXiv for recent and foundational papers on Newman's theorem in communication complexity. In communication complexity, Newman’s theorem is the statement that public randomness can be simulated by private randomness with only a logarithmic additive increase in communication. For a Boolean function $f:\{0,1\}^n\times\{0,1\}^n\to\{0,1\}$ and parameters $\epsilon,\delta>0$, the theorem is stated as

$$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$

so a public-coin protocol of error at most $\epsilon$ can be converted into a private-coin protocol whose error is at most $\epsilon+\delta$, with only $O(\log n+\log \delta^{-1})$ additional communication (Li et al., 2024).

1. Statement and role in randomized communication complexity

For a Boolean function

$f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$

the standard randomized communication complexity parameters are:

• $R^{pub}_{\epsilon}(f)$: the minimum communication cost of a protocol that computes $f$ with public coins and error at most $\epsilon$.
• $\epsilon,\delta>0$0: the analogous quantity with private coins.

Newman’s theorem asserts that the gap between these two models is only logarithmic in the input length, up to an additional $\epsilon,\delta>0$1 term when one allows the error to increase from $\epsilon,\delta>0$2 to $\epsilon,\delta>0$3. In the formulation above, the theorem is exactly the usual statement that public randomness only helps by a logarithmic amount.

This theorem is fundamental because it identifies public coins as a limited resource in the two-party model. It does not eliminate the distinction between public-coin and private-coin protocols, but it shows that the asymptotic advantage of public randomness is tightly controlled.

2. Geometric representation of protocols

The proof framework in (Li et al., 2024) places the problem in a finite-dimensional normed space. The function table of $\epsilon,\delta>0$4 is viewed as a vector

$\epsilon,\delta>0$5

with coordinates indexed by input pairs $\epsilon,\delta>0$6. The norm used throughout is the $\epsilon,\delta>0$7-norm,

$\epsilon,\delta>0$8

A public-coin protocol is interpreted as a convex combination of deterministic protocols. If the public randomness chooses deterministic protocol $\epsilon,\delta>0$9 with probability $$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$0, then the protocol is written as

$$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$1

where $$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$2 is the set of deterministic protocols in the support.

Two approximation relations organize the argument: $$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$3 and, after compression,

$$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$4

By the triangle inequality,

$$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$5

This reformulation converts the theorem into a problem about approximating a point in a convex hull by a sparse convex combination while controlling $$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$6-error.

3. Carathéodory-type ingredients

The streamlined proof relies on two distinct Carathéodory statements.

First, the classical Carathéodory theorem: if $$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$7, then every $$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$8 can be written as a convex combination of at most $$R^{pri}_{\epsilon+\delta}(f)\le R^{pub}_{\epsilon}(f)+O(\log n+\log \delta^{-1}),$$9 points of $\epsilon$0.

Second, the approximate Carathéodory theorem: if $\epsilon$1 satisfies

$\epsilon$2

then for every $\epsilon$3 and every $\epsilon$4, there exists $\epsilon$5 such that

$\epsilon$6

and $\epsilon$7 is a convex combination of only

$\epsilon$8

points from $\epsilon$9.

Within the proof, these two theorems play different roles. The classical theorem first reduces the ambient support to a dimension-dependent set. The approximate theorem then sparsifies the representation quantitatively, with explicit dependence on $\epsilon+\delta$0 and $\epsilon+\delta$1.

4. The Carathéodory proof of Newman’s theorem

The argument starts from a public-coin protocol of cost $\epsilon+\delta$2, represented as

$\epsilon+\delta$3

for some set $\epsilon+\delta$4 of deterministic protocols.

The first step is support compression via classical Carathéodory. Since $\epsilon+\delta$5, there exists a subset $\epsilon+\delta$6 such that

$\epsilon+\delta$7

and still

$\epsilon+\delta$8

The second step is sparse approximation via approximate Carathéodory. Each deterministic protocol $\epsilon+\delta$9 is Boolean-valued, so

$O(\log n+\log \delta^{-1})$0

Applying the approximate theorem yields a point $O(\log n+\log \delta^{-1})$1 satisfying

$O(\log n+\log \delta^{-1})$2

where $O(\log n+\log \delta^{-1})$3 is a convex combination of only

$O(\log n+\log \delta^{-1})$4

points from $O(\log n+\log \delta^{-1})$5.

The third step is the conversion to a private-coin protocol. Writing

$O(\log n+\log \delta^{-1})$6

Alice privately samples an index $O(\log n+\log \delta^{-1})$7 according to $O(\log n+\log \delta^{-1})$8, sends $O(\log n+\log \delta^{-1})$9 to Bob, and they run the deterministic protocol $f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$0. Specifying $f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$1 costs at most

$f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$2

bits, so the extra communication is

$f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$3

Finally, because

$f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$4

one has

$f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$5

and therefore the additional communication is

$f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$6

Combining this with

$f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$7

gives

$f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$8

which is the desired private-coin simulation.

5. Quantitative content

The proof is organized around a small set of central inequalities and sparsity bounds: $f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\},$9

$R^{pub}_{\epsilon}(f)$0

$R^{pub}_{\epsilon}(f)$1

$R^{pub}_{\epsilon}(f)$2

$R^{pub}_{\epsilon}(f)$3

and

$R^{pub}_{\epsilon}(f)$4

These formulas exhibit the structure of the theorem precisely. The public-coin protocol is first reduced to a convex combination supported on at most $R^{pub}_{\epsilon}(f)$5 deterministic protocols, then approximated by a combination of only $R^{pub}_{\epsilon}(f)$6 protocols, and finally implemented privately by sending the sampled protocol index.

A notable aspect of this bound is that the overhead depends on $R^{pub}_{\epsilon}(f)$7 only through $R^{pub}_{\epsilon}(f)$8, while the accuracy parameter enters through $R^{pub}_{\epsilon}(f)$9.

6. Relation to standard proofs and conceptual significance

The 2024 paper “Newman’s theorem via Carathéodory” emphasizes that this proof is shorter and cleaner than the standard probabilistic-method/Chernoff-bound proof (Li et al., 2024). In the standard argument, one samples a small multiset of public random strings and uses a Chernoff bound together with a union bound to show that the sampled randomness approximates the public-coin protocol on all inputs.

The Carathéodory-based proof replaces that sampling analysis with a geometric compression argument. A public-coin protocol is treated as a point in a convex hull of deterministic protocols; classical Carathéodory reduces the relevant support, and approximate Carathéodory sparsifies it further while controlling the $f$0-error. The resulting proof avoids explicit probabilistic sampling calculations and recasts the theorem as a statement about sparse convex approximation.

This geometric viewpoint isolates the mechanism behind the theorem: public randomness is a convex combination over deterministic protocols, and Carathéodory-type theorems show that such combinations can be compressed to small support without losing much in $f$1-accuracy. In that sense, Newman’s theorem becomes a direct statement about convex-hull sparsification in the communication-complexity model.