# The privacy loss random variable

This post is part of a series on differential privacy. Check out the table of contents to see the other articles!

R**emember** the notion of « almost »
differential privacy? We changed the original definition to add a new
parameter, \(\delta\). We said that \(\delta\) was « the probability that something
goes wrong ». This was a bit of a shortcut: this nice and easy intuition is
sometimes not exactly accurate. In this post, I'll do two things. I'll introduce
a crucial concept in differential privacy: the « privacy loss random variable ».
Then, I'll use it to explain what \(\delta\) *really* means.

Friendly heads-up: this post has slightly more math than the rest of this series. But don't worry! I made it as nice and visual as I could, with graphs instead of equations. All the equations are in a proof hidden by default.

# The privacy loss random variable

Recall the setting of the definition of \(\varepsilon\)-DP (short for differential privacy). The attacker tries to distinguish between two databases \(D_1\) and \(D_2\), that differ by only one record. If a mechanism \(A\) is \(\varepsilon\)-DP, then \(A\left(D_1\right)\) and \(A\left(D_2\right)\) will return output \(O\) with similar probability:

The equality also goes in the other direction, but the relation between \(D_1\) and \(D_2\) is symmetrical, so we only use this one inequality, to simplify.

We said before that the \(\varepsilon\) in \(\varepsilon\)-DP was the *maximal
knowledge gain* of the attacker. We defined this knowledge gain in Bayesian
terms, where the attacker is trying to guess if the real database \(D\) is \(D_1\)
and \(D_2\). We saw that \(\varepsilon\) bounds the evolution of betting odds. For
each \(O\), we had:

What if we don't just want to bound this quantity, but calculate it for a given output \(O\)? Let us define:

This formula looks scary, but the intuition behind it is pretty simple. The
denominator corresponds to the *initial* betting odds for \(D_1\) vs. \(D_2\). How
likely is one option vs. the other, before looking at the result of the
mechanism. In Bayesian terms, this is called the "prior". Meanwhile, the
numerator of the fraction is the betting odds *afterwards* — the "posterior".
Differential privacy guarantees that \(\mathcal{L}_{D_1,D_2}(O)\le\varepsilon\)
for all \(O\).

Bayes' rule allows us to reformulate this quantity:

This is called the *privacy loss random variable* (PLRV for short).
**Intuitively, the PLRV is the « actual \(\varepsilon\) value » for a specific
output \(O\).** Why is it a random variable? Because typically, we consider
\(\mathcal{L}_{D_1,D_2}(O)\) when \(O\) varies according to \(A(D_1)\), which we
assume is the "real" database.

OK, this is very abstract. We need a concrete example.

# A concrete example

Suppose that we're counting the number of people with blue eyes in the dataset.
We make this diferentially private by adding Laplace noise of scale
\(3\), to get \(\varepsilon=\ln(3)\). The attacker hesitates between two possible
datasets: one with \(1000\) blue-eyed people, the other with \(1001\). The *real*
number is \(1000\), but the attacker doesn't know that. The two distributions look
like this:

Let's consider three possible outputs of the mechanism, given the "real" database is \(D_1\). We represent them below as \(O_1\), \(O_2\), and \(O_3\).

Say the attacker is very uncertain: initially, they give equal probabilities to \(D_1\) and \(D_2\). What are they going to think once we give them the output of the mechanism?

- If we return \(O_1\), the attacker is starting to suspect that the real database is \(D_1\). There's a larger chance to get that output if \(D=D_1\) than if \(D=D_2\). How much larger? Exactly 3 times larger: the attacker's knowledge is tripled.
- If we return \(O_2\), the attacker is like: ¯\_(ツ)_/¯. This is not giving them much information. This output could have come from \(D_1\), but it could just as well have come from \(D_2\). The attacker's knowledge doesn't change.
- If we return \(O_3\), the attacker is getting
*tricked*with wrong information. They will think it's more likely that the real database is \(D_2\). Their "knowledge" is divided by 3.

Let's look at all possible events \(O=A(D_1)\), and *order* them. We'll put the
ones that help the attacker most first, and look at the value of
\(\mathcal{L}_{D_1,D_2}(O)\). Let's call this \(\mathcal{L}\), for short, and plot
it.

This is why Laplace noise is so nice: look at this neat horizontal line. Oh my god. It even has a straight diagonal. It never goes above \(\varepsilon\approx1.1\): a beautiful visual proof that Laplace noise gives \(\varepsilon\)-DP.

Let's change the graph above to more accurately represent that \(\mathcal{L}\) is
a *random variable*. On the \(x\)-axis, we represent all events according to their
probability. We're also more interested in \(\exp(\mathcal{L})\), so let's plot
that instead of \(\mathcal{L}\).

Now, what if you were using some other type of noise? Say, from a normal distribution? It would make data analysts happier: Laplace noise is weird to them, it never shows up in the real world. Normal distributions, by contrast, are familiar and friendly. A lot of natural data distributions can be modeled with them.

In the context of differential privacy, the normal distribution is called « Gaussian noise ». Let's try to add Gaussian noise, of variance \(\sigma^2=2\):

OK, looks reasonable, now let's see what \(e^\mathcal{L}\) looks like:

Ew. Look at this line going up to infinity on the left side. Gross. We can't just draw a line at \(e^\varepsilon\) and say "everything is underneath". What do we do, then? We cheat, and use a \(\delta\).

# \(\delta\) and the PLRV

In a previous article, we said that the \(\delta\) in \((\varepsilon,\delta)\)-DP is the probability that something terrible happens. What does that mean in the context of Gaussian noise? First, we pick an arbitrary \(\varepsilon\), say, \(\varepsilon=\ln(3)\). Then, we look at how likely it for \(e^\mathcal{L}\) to be above the \(e^\varepsilon=3\) line. It's easy to do: the \(x\)-axis is the probability space, so we can simply measure the width of the bad events.

This simple intuition is correct: this mechanism is \((\ln(3),\delta_1)\)-DP, with \(\delta_1\approx0.054\). But it misses an important subtlety. Let's zoom in on the part where things go wrong, and consider two possible outputs.

Returning \(O_1\) is not great: \(e^\mathcal{L}>e^\varepsilon\). But it's not
*terrible*: the privacy loss is only a tiny bit larger than we'd hope. Returning
\(O_2\), however, is scary news: \(e^\mathcal{L}\) is huge. Intuitively, \(O_2\) leaks
much more information than \(O_1\).

With our way of quantifying \(\delta\), we don't account for this. We only measure
the \(x\)-axis. What we count is *whether* \(e^\mathcal{L}\) is above the line, not
*how much* it's above the line. For each bad event of probability \(p\), we're
adding \(p\times1\) to the \(\delta\). A finer approach is to *weigh* the bad events
by "how bad they are". We want to give a "weight" of \(\approx1\) to the very bad
events, and a weight of \(\approx0\) to the "not too bad" ones.

To do this, we transform a bit the curve above by doing two things. First, we
take the *inverse* of the curve: very bad events are now close to \(0\) instead of
very large. Second, we *normalize* the curve by taking the ratio
\(e^\varepsilon/e^\mathcal{L}\). This way, events that are "not too bad" are close
to \(1\).

This allows us to consider the *area* between the curve and the \(y=1\) line. When
\(\mathcal{L}\) is very large, the inverse is close to \(0\), so the distance to \(1\)
is almost 1. And when \(\mathcal{L}\) is close to \(\varepsilon\), the ratio is one,
and the distance is almost 0. *Very bad* events count more than *sort of bad*
events.

This is the tighter, exact characterization of \(\delta\). In
\((\varepsilon,\delta)\)-DP, the \(\delta\) is the area highlighted above. It is the
mass of all possible bad events, *weighted* by how likely they are and how bad
they are. This tells us that the mechanism is \((\ln(3),\delta_2)\)-DP with
\(\delta_2\approx0.011\), a much better characterization than before.

The typical definition of \((\varepsilon,\delta)\)-DP doesn't use this complicated formulation. A mechanism \(A\) is \((\varepsilon,\delta)\)-DP if for any neighboring \(D_1\) and \(D_2\), and any set \(S\) of possible outputs:

This definition is equivalent to the previous characterization. If you want to see the proof of that, click here:

# What about infinity values?

Using Gaussian noise, all possible values of \(\mathcal{L}\) are *finite*. But for
some mechanisms \(A\), there are outputs \(O\) such that \(\mathbb{P}[A(D_1)=O]>0\),
but \(\mathbb{P}[A(D_2)=O]=0\). In that case, \(\mathcal{L}(O)=\infty\). This kind
of output is called a *distinguishing event*. If we return a distinguishing
event, the attacker immediately finds out that \(D\) is \(D_1\) and not \(D_2\). This
is the case for the "thresholding" example we looked at previously.

Our interpretation of \(\delta\) captures this nicely. Since we inverted the curve, if \(\mathcal{L}=\infty\), we simply have \(e^\varepsilon/e^\mathcal{L}=0\). The distance to \(1\) is exactly \(1\), so we count these events with maximal weight. The graph looks like this:

In that case, \(\delta_1=\delta_2\): all "bad" events are worst-case events. For such a mechanism, the two characterizations of \(\delta\) are the same.

# Final note

You might be wondering: why use Gaussian noise at all if it requires \(\delta>0\)?

This is an excellent question. I'm glad you asked it, because it is exactly the topic of the next blog post in this series. Or you can, as always, select another article to read next in the table of contents!

Thanks to Sebastian Meiser, who wrote the reference paper about the subtleties with \(\delta\). It makes for excellent reading if you want to dig a bit deeper into this. Thanks also to Antoine Amarilli for proofreading this blog post.