---
title: Newman's Theorem in Communication Complexity
url: https://www.emergentmind.com/topics/newman-s-theorem
type: topic
---

# Newman's Theorem in Communication Complexity

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 [2406.08500].

## 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 \(\epsilon\).
- \(R^{pri}_{\epsilon}(f)\): 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 \(\log \delta^{-1}\) term when one allows the error to increase from \(\epsilon\) to \(\epsilon+\delta\). 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 [2406.08500] places the problem in a finite-dimensional normed space. The function table of \(f\) is viewed as a vector
\[
f\in \mathbb{R}^N,\qquad N=2^{n+1},
\]
with coordinates indexed by input pairs \((x,y)\). The norm used throughout is the \(L_\infty\)-norm,
\[
\|z\|_\infty=\max_i |z_i|.
\]

A public-coin protocol is interpreted as a convex combination of deterministic protocols. If the public randomness chooses deterministic protocol \(P_i\) with probability \(\mu_i\), then the protocol is written as
\[
P=\sum_i \mu_i P_i \in \operatorname{conv}(S),
\]
where \(S=\{P_1,\dots,P_q\}\) is the set of deterministic protocols in the support.

Two approximation relations organize the argument:
\[
\|f-P\|_\infty \le \epsilon,
\]
and, after compression,
\[
\|P-P'\|_\infty \le \delta.
\]
By the triangle inequality,
\[
\|f-P'\|_\infty \le \epsilon+\delta.
\]

This reformulation converts the theorem into a problem about approximating a point in a convex hull by a sparse convex combination while controlling \(L_\infty\)-error.

## 3. Carathéodory-type ingredients

The streamlined proof relies on two distinct Carathéodory statements.

First, the **classical Carathéodory theorem**: if \(S\subseteq \mathbb{R}^n\), then every \(x\in \operatorname{conv}(S)\) can be written as a convex combination of at most \(n+1\) points of \(S\).

Second, the **approximate Carathéodory theorem**: if \(S\subseteq\mathbb{R}^n\) satisfies
\[
\max_{y\in S}\|y\|_\infty \le 1,
\]
then for every \(x\in \operatorname{conv}(S)\) and every \(\delta>0\), there exists \(x'\in \operatorname{conv}(S)\) such that
\[
\|x-x'\|_\infty \le \delta,
\]
and \(x'\) is a convex combination of only
\[
O\!\left(\frac{\log |S|}{\delta}\right)
\]
points from \(S\).

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 \(\log |S|\) and \(\delta^{-1}\).

## 4. The Carathéodory proof of Newman’s theorem

The argument starts from a public-coin protocol of cost \(R^{pub}_{\epsilon}(f)\), represented as
\[
P\in \operatorname{conv}(S)
\]
for some set \(S=\{P_1,\dots,P_q\}\) of deterministic protocols.

The first step is **support compression via classical Carathéodory**. Since \(P\in \operatorname{conv}(S)\subseteq \mathbb{R}^N\), there exists a subset \(S'\subseteq S\) such that
\[
|S'|\le N+1
\]
and still
\[
P\in \operatorname{conv}(S').
\]

The second step is **sparse approximation via approximate Carathéodory**. Each deterministic protocol \(P_i\) is Boolean-valued, so
\[
\max_{y\in S'}\|y\|_\infty \le 1.
\]
Applying the approximate theorem yields a point \(P'\in \operatorname{conv}(S')\) satisfying
\[
\|P-P'\|_\infty \le \delta,
\]
where \(P'\) is a convex combination of only
\[
k=O\!\left(\frac{\log |S'|}{\delta}\right)
\]
points from \(S'\).

The third step is the **conversion to a private-coin protocol**. Writing
\[
P'=\sum_{i=1}^k \mu_i P_i,\qquad \sum_i \mu_i=1,\ \mu_i\in[0,1],
\]
Alice privately samples an index \(i\in[k]\) according to \((\mu_1,\dots,\mu_k)\), sends \(i\) to Bob, and they run the deterministic protocol \(P_i\). Specifying \(i\) costs at most
\[
\log k+1
\]
bits, so the extra communication is
\[
O(\log\log |S'|+\log \delta^{-1}).
\]

Finally, because
\[
|S'|\le N+1=2^{n+1}+1,
\]
one has
\[
\log\log |S'|=O(\log n),
\]
and therefore the additional communication is
\[
O(\log n+\log \delta^{-1}).
\]
Combining this with
\[
\|f-P\|_\infty \le \epsilon
\quad\text{and}\quad
\|P-P'\|_\infty \le \delta
\]
gives
\[
\|f-P'\|_\infty \le \epsilon+\delta,
\]
which is the desired private-coin simulation.

## 5. Quantitative content

The proof is organized around a small set of central inequalities and sparsity bounds:
\[
\|f-P\|_\infty \le \epsilon,
\]
\[
\|P-P'\|_\infty \le \delta,
\]
\[
\|f-P'\|_\infty \le \epsilon+\delta,
\]
\[
|S'|\le N+1,\qquad N=2^{n+1},
\]
\[
k = O\!\left(\frac{\log |S'|}{\delta}\right),
\]
and
\[
\log k + 1 = O(\log\log |S'|+\log \delta^{-1}) = O(\log n+\log \delta^{-1}).
\]

These formulas exhibit the structure of the theorem precisely. The public-coin protocol is first reduced to a convex combination supported on at most \(N+1\) deterministic protocols, then approximated by a combination of only \(k\) protocols, and finally implemented privately by sending the sampled protocol index.

A notable aspect of this bound is that the overhead depends on \(n\) only through \(\log n\), while the accuracy parameter enters through \(\log \delta^{-1}\).

## 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 [2406.08500]. 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 \(L_\infty\)-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 \(L_\infty\)-accuracy. In that sense, Newman’s theorem becomes a direct statement about convex-hull sparsification in the communication-complexity model.

Source: https://www.emergentmind.com/topics/newman-s-theorem