Lecture 8: Learning Juntas

Topics: Junta testing, Junta learning, Relevant variables

References: [Mossel-O'Donnell-Servedio'04], [Valiant'12]
Based on lecture notes by: Timothe Kasriel
Notebook by: Gabriel Taboada

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Key Concepts

Term	Definition
$k$-Junta	Function depending on at most $k$ variables
Relevant variable	Variable $i$ where $\text{Inf}_i[f] > 0$
Junta Testing	Determine if $f$ is close to a $k$-junta
Junta Learning	Learn which $k$ variables matter

Key insight: A function is a $k$-junta iff it has at most $k$ relevant variables.

# Install/upgrade boofun (required for Colab)
!pip install --upgrade boofun -q

import boofun as bf
print(f"BooFun version: {bf.__version__}")

[notice] A new release of pip is available: 25.2 -> 26.0
[notice] To update, run: /Library/Developer/CommandLineTools/usr/bin/python3 -m pip install --upgrade pip

/Users/gabrieltaboada/dev/Boofun/boofun/src/boofun/core/errormodels.py:21: UserWarning: uncertainties library not available - some error models disabled
  warnings.warn("uncertainties library not available - some error models disabled")

/Users/gabrieltaboada/dev/Boofun/boofun/src/boofun/quantum/__init__.py:22: UserWarning: Qiskit not available - quantum features limited
  warnings.warn("Qiskit not available - quantum features limited")

BooFun version: 1.1.1

import numpy as np
import boofun as bf

# Example functions
dictator = bf.dictator(8, 0)  # f(x) = x_0 (1 relevant variable)
maj7 = bf.majority(7)         # Majority on 7 variables (ALL 7 relevant)
and4 = bf.AND(4)              # AND of 4 variables (all 4 relevant)

print("Testing k-junta properties:")
print("="*55)

# Dictator: only x_0 matters → 1-junta
print(f"\nDictator-8: f(x) = x_0")
print(f"  Influences: {[f'{inf:.2f}' for inf in dictator.influences()]}")
print(f"  is_junta(1): {dictator.is_junta(1)}")

# Majority-7: ALL variables matter → exactly 7-junta
print(f"\nMajority-7: f(x) = sign(x_0 + ... + x_6)")
print(f"  Influences: {[f'{inf:.2f}' for inf in maj7.influences()]}")
print(f"  is_junta(3): {maj7.is_junta(3)}  ← NOT a 3-junta (needs all 7!)")
print(f"  is_junta(6): {maj7.is_junta(6)}  ← NOT a 6-junta")
print(f"  is_junta(7): {maj7.is_junta(7)}  ← IS a 7-junta")

# AND-4: all variables matter
print(f"\nAND-4: f(x) = x_0 ∧ x_1 ∧ x_2 ∧ x_3")
print(f"  Influences: {[f'{inf:.2f}' for inf in and4.influences()]}")
print(f"  is_junta(3): {and4.is_junta(3)}")
print(f"  is_junta(4): {and4.is_junta(4)}")

Testing k-junta properties:
=======================================================

Dictator-8: f(x) = x_0
  Influences: ['1.00', '0.00', '0.00', '0.00', '0.00', '0.00', '0.00', '0.00']
  is_junta(1): True

Majority-7: f(x) = sign(x_0 + ... + x_6)
  Influences: ['0.31', '0.31', '0.31', '0.31', '0.31', '0.31', '0.31']
  is_junta(3): False  ← NOT a 3-junta (needs all 7!)
  is_junta(6): False  ← NOT a 6-junta
  is_junta(7): True  ← IS a 7-junta

AND-4: f(x) = x_0 ∧ x_1 ∧ x_2 ∧ x_3
  Influences: ['0.12', '0.12', '0.12', '0.12']
  is_junta(3): False
  is_junta(4): True

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Summary

• $k$-Junta: Depends on at most $k$ variables (has ≤ $k$ relevant variables)
• Test: Check if total influence on non-junta variables is small
• Learn: Use Goldreich-Levin to find relevant variables via heavy Fourier coefficients

Runtime ([MOS04]): $\approx n^{k\omega/(\omega+1)}$ where $\omega \approx 2.37$ is matrix multiplication exponent

f.is_junta(k)        # Check if f is a k-junta
f.influences()       # Per-variable influences
f.total_influence()  # Sum of all influences