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Model

Differential-privacy noise and budget

DP is nearly free on large aggregates and very costly on small ones; and some budgets are depleted by reading.

Privacy parameter
budget spent by one release (smaller is more private)
Sensitivity
the most that one individual can change the answer (1 for a count)

Trade-off: Exactness vs. approximation

How it behaves

One input on the horizontal axis, the others held fixed.

ε = 0.5ε = 2
Data table
Typical relative error of a noisy count, by true count
True countseriesRelative error
1ε = 0.5200%
1.26ε = 0.5159%
1.58ε = 0.5126%
2ε = 0.5100%
2.51ε = 0.579.6%
3.16ε = 0.563.2%
3.98ε = 0.550.2%
5.01ε = 0.539.9%
6.31ε = 0.531.7%
7.94ε = 0.525.2%
10ε = 0.520%
12.6ε = 0.515.9%
15.8ε = 0.512.6%
20ε = 0.510%
25.1ε = 0.57.96%
31.6ε = 0.56.32%
39.8ε = 0.55.02%
50.1ε = 0.53.99%
63.1ε = 0.53.17%
79.4ε = 0.52.52%
100ε = 0.52%
126ε = 0.51.59%
158ε = 0.51.26%
200ε = 0.51%
251ε = 0.50.796%
316ε = 0.50.632%
398ε = 0.50.502%
501ε = 0.50.399%
631ε = 0.50.317%
794ε = 0.50.252%
1Kε = 0.50.2%
1.26Kε = 0.50.159%
1.58Kε = 0.50.126%
2Kε = 0.50.1%
2.51Kε = 0.50.08%
3.16Kε = 0.50.063%
3.98Kε = 0.50.05%
5.01Kε = 0.50.04%
6.31Kε = 0.50.032%
7.94Kε = 0.50.025%
10Kε = 0.50.02%
12.6Kε = 0.50.016%
15.8Kε = 0.50.013%
20Kε = 0.50.01%
25.1Kε = 0.50.008%
31.6Kε = 0.50.006%
39.8Kε = 0.50.005%
50.1Kε = 0.50.004%
63.1Kε = 0.50.003%
79.4Kε = 0.50.003%
100Kε = 0.50.002%
126Kε = 0.50.002%
158Kε = 0.50.001%
200Kε = 0.50.001%
251Kε = 0.50.001%
316Kε = 0.50.001%
398Kε = 0.50.001%
501Kε = 0.50%
631Kε = 0.50%
794Kε = 0.50%
1Mε = 0.50%
1.26Mε = 0.50%
1.58Mε = 0.50%
2Mε = 0.50%
2.51Mε = 0.50%
3.16Mε = 0.50%
3.98Mε = 0.50%
5.01Mε = 0.50%
6.31Mε = 0.50%
7.94Mε = 0.50%
10Mε = 0.50%
12.6Mε = 0.50%
15.8Mε = 0.50%
20Mε = 0.50%
25.1Mε = 0.50%
31.6Mε = 0.50%
39.8Mε = 0.50%
50.1Mε = 0.50%
63.1Mε = 0.50%
79.4Mε = 0.50%
100Mε = 0.50%
1ε = 250%
1.26ε = 239.7%
1.58ε = 231.5%
2ε = 225.1%
2.51ε = 219.9%
3.16ε = 215.8%
3.98ε = 212.6%
5.01ε = 29.98%
6.31ε = 27.92%
7.94ε = 26.29%
10ε = 25%
12.6ε = 23.97%
15.8ε = 23.15%
20ε = 22.51%
25.1ε = 21.99%
31.6ε = 21.58%
39.8ε = 21.26%
50.1ε = 20.998%
63.1ε = 20.792%
79.4ε = 20.629%
100ε = 20.5%
126ε = 20.397%
158ε = 20.315%
200ε = 20.251%
251ε = 20.199%
316ε = 20.158%
398ε = 20.126%
501ε = 20.1%
631ε = 20.079%
794ε = 20.063%
1Kε = 20.05%
1.26Kε = 20.04%
1.58Kε = 20.032%
2Kε = 20.025%
2.51Kε = 20.02%
3.16Kε = 20.016%
3.98Kε = 20.013%
5.01Kε = 20.01%
6.31Kε = 20.008%
7.94Kε = 20.006%
10Kε = 20.005%
12.6Kε = 20.004%
15.8Kε = 20.003%
20Kε = 20.003%
25.1Kε = 20.002%
31.6Kε = 20.002%
39.8Kε = 20.001%
50.1Kε = 20.001%
63.1Kε = 20.001%
79.4Kε = 20.001%
100Kε = 20.001%
126Kε = 20%
158Kε = 20%
200Kε = 20%
251Kε = 20%
316Kε = 20%
398Kε = 20%
501Kε = 20%
631Kε = 20%
794Kε = 20%
1Mε = 20%
1.26Mε = 20%
1.58Mε = 20%
2Mε = 20%
2.51Mε = 20%
3.16Mε = 20%
3.98Mε = 20%
5.01Mε = 20%
6.31Mε = 20%
7.94Mε = 20%
10Mε = 20%
12.6Mε = 20%
15.8Mε = 20%
20Mε = 20%
25.1Mε = 20%
31.6Mε = 20%
39.8Mε = 20%
50.1Mε = 20%
63.1Mε = 20%
79.4Mε = 20%
100Mε = 20%

Worked examples, recomputed

Every number the mechanisms quote for this relationship is recomputed from it, so none of them disagrees with it.

Where it appears

Each mechanism states it in its own names. The general symbol each name stands for is shown with it.

Dictionary encoding: when it pays

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