Absolute–Mean Quantization Function

• Absolute–Mean Quantization Function is a randomized protocol that uses shared randomness and the empirical mean deviation to efficiently quantize data for distributed estimation.
• The method achieves a lower mean-squared error by scaling with the mean deviation rather than the full data range, outperforming traditional range-dependent schemes.
• Its low communication overhead and practical design make it a key subroutine in distributed optimization and aggregation tasks.

The absolute–mean quantization function, also known as the mean-deviation quantizer, is a randomized quantization protocol tailored for distributed mean estimation under communication constraints. Its characteristic distinction is that the leading term of its mean-squared error (MSE) bound depends on the empirical mean deviation of the data, $\sigma_{\rm m}$, rather than the full absolute range—a feature not achieved by earlier protocols without stringent assumptions. The quantizer is central to a correlated quantization scheme, which leverages shared randomness to attain optimal estimation error with minimal communication and without prior knowledge of data concentration properties (Suresh et al., 2022).

1. Mathematical Definition and Construction

Suppose $n$ clients each possess a real value $x_i \in [\ell, r]$. The global empirical mean and absolute mean deviation are given by

$$\bar x = \frac{1}{n}\sum_{i=1}^n x_i, \qquad \sigma_{\rm m} = \frac{1}{n}\sum_{i=1}^n |x_i - \bar x|.$$

The quantization process proceeds as follows:

• A public random permutation $\pi$ of $\{0, \ldots, n-1\}$, and independent shifts $\gamma_i \sim U[0,1/n)$, are fixed.
• Define $U_i = \pi_i/n + \gamma_i$ and scaled value $y_i = (x_i-\ell)/(r-\ell)$.
• A public random base $c_1 \sim U[-1/k,0)$ is chosen; quantization step size $\beta = \frac{k+1}{k(k-1)}$, and $c_j = c_1 + (j-1)\beta$ for $j=1,\ldots,k$. This partitions $[0,1]$ into $k$ overlapping levels.
• For each $x_i$, set $c'_i = \max\{c_j:\ c_j < y_i\}$, and define the quantized value as

$$Q_i(x_i) = (\ell - r) \Bigl[ c'_i + \beta \cdot 1_{\{U_i < y_i\}} \Bigr].$$

In higher-dimensional settings, this construction is applied coordinate-wise, or preceded by a random Hadamard rotation to improve $\ell_\infty$ norm performance.

2. Error Analysis and Optimality

For $k \ge 3$ quantization levels, the protocol’s estimator is $\hat x = \frac{1}{n} \sum_{i} Q_i(x_i)$. The mean squared error (MSE) satisfies

$$\mathbb{E}\|\hat x - \bar x\|_2^2 \leq \frac{12}{n} \min\left( \frac{\sigma_{\rm m}(r-\ell)}{k},\, \frac{(r-\ell)^2}{k^2} \right) + \frac{48 (r-\ell)^2}{n^2 k^2}.$$

Crucially, the leading term scales as $\sigma_{\rm m}/(n k)$, for arbitrary data concentration. This establishes preferable error decay when the $x_i$ are concentrated (i.e., small $\sigma_{\rm m}$), unlike range-dependent quantizers.

Variance analysis using sampling without replacement arguments yields these bounds. A matching lower bound (up to constants) via Yao’s principle shows no $k$-level interval quantizer can outperform $$\Omega(\sigma_{\rm m}/(n k) + (r-\ell)^2/(n^2 k^2))$$.

3. Protocol Description and Implementation

The quantization protocol employs only public shared randomness—specifically, a random permutation, independent shifts, and a base offset—each generated with $O(\log n)$ bits of server-seeded randomness. The protocol's essential steps are outlined as follows:

Step	Operation	Notes
Randomness	Choose $\pi$, $\gamma_i$, $c_1$	All public and seedable
Client Step	Compute $y_i$, $U_i$, find $c'_i$, quantize $x_i$	Output $Q_i(x_i)$; $k$ possible codes
Server	Aggregate $\hat{x}$ as scaled mean of quantized values

No assumption on the size or prior knowledge of $\sigma_{\rm m}$ is required; the protocol implicitly estimates $\sigma_{\rm m}$ via randomness structures.

4. Comparison: Correlated vs. Independent Schemes

A canonical case with $n = 2$, $\ell = 0$, $r = 1$, and binary quantization ($k=2$), demonstrates the efficacy of correlation. If each client sends $Q_i(x_i) = 1_{U_i < x_i}$ with $U_1 \sim U[0,1]$, $U_2 = 1-U_1$:

• For $x_1 = x_2 = x$, the independent scheme has MSE $\frac{1}{2} x(1-x)$,
• The correlated scheme achieves $\frac{x}{2} + \max\{x-\frac{1}{2},0\} - x^2$, always less than or equal to the independent case, and exactly zero at $x \in \{0, 1/2, 1\}$.

This illustrates the strict advantage in settings with minimal mean deviation.

5. Assumptions, Information Requirements, and Practical Use

The only assumption mandated is $x_i \in [\ell, r]$ for all $i$; no data-dependent initialization is needed. The protocol’s information overhead is minimal and fully public. In practice, this method can be utilized as a subroutine in distributed optimization, yielding improved convergence rates over prior protocols whose error terms depend on the absolute range or require dataset concentration estimates. Experimental evidence demonstrates performance advantages on diverse tasks (Suresh et al., 2022).

6. Theoretical and Empirical Impact

The absolute–mean quantization function, as formalized by Suresh, Sun, Ro, and Yu (Google Research, 2023), establishes a new performance benchmark for distributed mean estimation and distributed optimization tasks, matching information-theoretic lower bounds up to constant factors in both error and communication. The protocol's dependency on mean deviation, rather than data range, obviates the need for heavy data concentration assumptions and motivates its utility in heterogeneous distributed systems. This quantizer is now a canonical baseline for analyzing quantized, communication-constrained aggregation (Suresh et al., 2022).