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Estimation with Quantized Parameter Side-Information
Mathis Wetterwald, Ofer Shayevitz, Michèle Wigger
TL;DR
The paper asks how to estimate a parameter when an estimator has both i.i.d. observations and a limited-level quantized version of the parameter. It analyzes optimized helper quantization and states that minimax risk still improves roughly by 1/k², with upper bounds improving on simple strategies for moderate or large sample and message counts.
Problem
The paper studies minimax parameter estimation when the estimator receives i.i.d. samples and a k-level quantized version of the unknown parameter, with the quantizer chosen to help estimation.
Source: §I · p. 1Method
The paper characterizes minimax risk by deriving converse and achievability bounds for estimators combining samples with a helper's quantized parameter message.
Source: §II · p. 2Results
The minimax risk still improves roughly by 1/k², and the upper bounds strictly improve on simple strategies for moderate or large k and n.
Source: §I · p. 1, §II · p. 2Takeaways & Limitations
The stated bounds combine the benefits of helper information and observations, whose separate simple strategies yield risks of order 1/k² and 1/n, respectively.
Source: §II · p. 2
Abstract
from arXiv · showhide
This paper presents upper and lower bounds on the minimax risk of a parameter estimation problem where the estimator not only observes independent and identically distributed observations but also a quantized version of the parameter, where the quantization has a limitation on the number of levels, but can be optimized otherwise. Our upper and lower bounds have similar decay rates for large number of observation samples and quantization levels.
I. INTRODUCTION
The paper studies minimax parameter estimation when an estimator receives both i.i.d. samples and a quantized version of the parameter. It finds that the minimax risk still improves roughly by 1/k², with an essentially optimal quantizer that is cyclically uniform at a scale tied to the no-quantizer estimation error.
I. INTRODUCTION
- The estimator observes n i.i.d. samples from a known parametric family and seeks an estimate minimizing worst-case risk over the parameter interval.The paper focuses on the minimax setting among the estimation paradigms described.Source: §I · p. 1
- The paper optimizes a k-level quantizer of the parameter, available to the estimator before it sees the samples, and studies its effect on minimax risk.Source: §I · p. 1
- The minimax risk still improves roughly by a factor of 1/k², using an essentially cyclically uniform quantizer whose cycle corresponds to the standard deviation of the no-quantizer estimation error.The introduction contrasts this with the uniform quantizer optimal when there are no samples.Source: §I · p. 1
- Related distributed-estimation work studies remote parties with samples governed by a shared parameter, whereas this paper considers a helper that directly accesses and encodes the parameter for a local estimator.Source: §I · p. 1
II. PROBLEM SETUP AND MAIN RESULT
The setup gives a perfectly informed helper a limited message alphabet and lets an estimator combine that message with i.i.d. observations. The paper states converse and achievability bounds, including a Bernoulli case, and says the upper bounds improve on simple helper-only or data-only strategies for moderate or large sample and message counts.
II. PROBLEM SETUP AND MAIN RESULT
- A perfectly informed helper sends a k-valued message, and the estimator combines it with n i.i.d. observations from Pθ to estimate θ.The parameter lies in a bounded interval and each Pθ is a pmf on a finite alphabet.Source: §II · pp. 1–2
- The objective is the best possible Bayes risk over helper strategies and estimators.Source: §II · p. 2
- The section states a converse under Assumption 1 and an achievability bound under Assumptions 1 and 2, with sufficiently large-sample conditions.For the Bernoulli-θ model on [0, 1], it also states a result when k > 2.Source: §II · p. 2
- Using only observations gives minimax risk of order 1/n, while partitioning the parameter interval equally gives risk 1/k² when ignoring the data.Source: §II · p. 2
- For moderate or large k and n, the achievability upper bounds strictly improve on those simple strategies.Source: §II · p. 2
A. Scheme
The helper partitions the parameter interval into subintervals and the estimator uses the helper message together with the observed sequence to choose a midpoint. Numerical evaluation suggests the resulting minimax risk decays approximately as 1/k² and 1/n.
A. Scheme
- The helper partitions the parameter interval into disjoint subintervals of length L, with a possibly shorter final subinterval.The paper proposes a helper-and-estimator strategy based on these intervals.Source: A. Scheme · p. 2
- Given the observations and helper message, the estimator selects the indicated midpoint closest to the empirical type.If the minimizer is not unique, it chooses one at random.Source: A. Scheme · p. 2
- For the Bernoulli-θ result, the paper modifies the choice of L from the general scheme.The numerical evaluation uses J = [0, 1] and Pθ = (θ, θ − θ², (1 − θ)²).Source: A. Scheme · p. 2
- The numerical evaluation suggests minimax risk is nearly proportional to 1/k² and decays proportionally to 1/n.The reported 1/k² decay is a little slower than the lower-bound rate, with the discrepancy narrowing as n increases.Source: A. Scheme · p. 2
B. Analysis
The analysis bounds the proposed strategy's expected mean squared error by controlling how far the empirical type lies from the parameter and how errors accumulate across intervals. A separate refinement treats the Bernoulli-θ example.
B. Analysis
- For the Bernoulli-θ example, the analysis modifies L, with the modified length positive for k > 2.In this binary example, δ-typicality is equivalent to |π_Xn(1) − θ| ≤ δ.Source: B. Analysis · p. 3
- The upper-bound argument controls squared error by considering intervals at increasing distances from the interval containing θ.It bounds probabilities using tail estimates and notes that the resulting infinite sum converges when e^(-nL²k²) < 1.Source: B. Analysis · pp. 3–4
- The analysis obtains a bound of 3/k² for the indicated error term.The passage states this bound after identifying the correct interval containing θ.Source: B. Analysis · pp. 3–4
IV. CONVERSE PROOF (PROOF OF THEOREM 1)
The converse proof restricts attention to a short parameter interval and a corresponding region of types. It then uses a modified two-point Le Cam method on that region.
IV. CONVERSE PROOF (PROOF OF THEOREM 1)
- For any subinterval I of parameter space J of length n^-1/2, the text defines R_I and states a lemma about its n-types for sufficiently large n and θ′ ∈ I, π ∈ R_I.Source: §IV · p. 4
- Because the helper function has only k possible values, parameter values in I of length 1/√n must exist that share a helper value.The proof uses such a pair in the restricted lower-bound argument.Source: §IV · p. 4
- The proof uses a modified two-point Le Cam method restricted to the region R_I.It selects two parameters in I that share a helper message.Source: §IV · p. 4
APPENDIX A PROOF OF LEMMA 1
The appendix proves Lemma 1 using type-count bounds and probability estimates. Its steps invoke Stirling’s and Herbert Robbins’ bounds, the arithmetic–geometric mean inequality, and Assumption 1.
APPENDIX A PROOF OF LEMMA 1
- Under Assumption (38), the number of n-types in the specified ball is lower bounded by (r/(m − 1))^(m − 1).Setting r = n^-1/2 is stated to conclude part 1 of the lemma.Source: Appendix A · p. 4
- The proof uses Stirling’s and Herbert Robbins’ bounds in its estimates.The supplied passage does not state the resulting expression.Source: Appendix A · p. 4
- The proof also uses the arithmetic–geometric mean inequality and the triangle inequality with Assumption 1.It considers θ in I and π in R_I, with θ′ chosen so that π lies in the 1/√n ball around Pθ′.Source: Appendix A · p. 5