Use better group privacy bound in computing user level privacy [TF Privacy]

PiperOrigin-RevId: 526852999

tensorflow_privacy/privacy/analysis/compute_dp_sgd_privacy_lib.py

@@ -28,6 +28,11 @@ class UserLevelDPComputationError(Exception):
   """Error raised if user-level epsilon computation fails."""
 
 
+def _logexpm1(x: float) -> float:
+  """Returns log(exp(x) - 1)."""
+  return x + math.log(-math.expm1(-x))
+
+
 def _compute_dp_sgd_user_privacy(
     num_epochs: float,
     noise_multiplier: float,
@@ -100,21 +105,32 @@ def _compute_dp_sgd_user_privacy(
   #   log(G(F(example_delta), example_delta)) - log(user_delta) = 0
   # Then we can return user_eps = H(F(example_delta)).
 
-  log_k = math.log(max_examples_per_user)
   target_user_log_delta = math.log(user_delta)
 
+  # We store all example_eps computed for any example_delta in the following
+  # method. This is done so that we don't have to recompute values for the same
+  # delta.
+  epsilon_cache = dict()
+
   def user_log_delta_gap(example_log_delta):
-    example_eps = _compute_dp_sgd_example_privacy(
+    if example_log_delta not in epsilon_cache:
+      epsilon_cache[example_log_delta] = _compute_dp_sgd_example_privacy(
           num_epochs,
           noise_multiplier,
           math.exp(example_log_delta),
           used_microbatching,
           poisson_subsampling_probability,
       )
+    example_eps = epsilon_cache[example_log_delta]
 
-    # Estimate user_eps, user_log_delta using Vadhan Lemma 2.2.
+    # Estimate user_eps, user_log_delta using Vadhan Lemma 2.2, using a tighter
+    # bound seen in the penultimate line of the proof, given as
+    # user_delta = (example_delta * (exp(k * example_eps) - 1)
+    #               / (exp(example_eps) - 1))
     user_eps = max_examples_per_user * example_eps
-    user_log_delta = log_k + user_eps + example_log_delta
+    user_log_delta = (
+        example_log_delta + _logexpm1(user_eps) - _logexpm1(example_eps)
+    )
     return user_log_delta - target_user_log_delta
 
   # We need bounds on the example-level delta. The supplied user-level delta
@@ -164,9 +180,16 @@ def _compute_dp_sgd_user_privacy(
     )
 
   # Vadhan (2017) "The complexity of differential privacy" Lemma 2.2.
-  # user_delta = k * exp(k * example_eps) * example_delta
+  if example_log_delta not in epsilon_cache:
-  # Given example_delta, we can solve for (k * example_eps) = user_eps.
+    epsilon_cache[example_log_delta] = _compute_dp_sgd_example_privacy(
-  return max(0, target_user_log_delta - log_k - example_log_delta)
+        num_epochs,
+        noise_multiplier,
+        math.exp(example_log_delta),
+        used_microbatching,
+        poisson_subsampling_probability,
+    )
+  example_eps = epsilon_cache[example_log_delta]
+  return max_examples_per_user * example_eps
 
 
 def _compute_dp_sgd_example_privacy(
@@ -354,7 +377,7 @@ using RDP accounting and group privacy:""",
     paragraphs.append(
         textwrap.fill(
             """\
-No user-level privacy guarantee is possible witout a bound on the number of \
+No user-level privacy guarantee is possible without a bound on the number of \
 examples per user.""",
             width=80,
         )

tensorflow_privacy/privacy/analysis/compute_dp_sgd_privacy_test.py

@@ -134,25 +134,35 @@ class ComputeDpSgdPrivacyTest(parameterized.TestCase):
     self.assertEqual(user_eps, example_eps)
 
   @parameterized.parameters((0.9, 2), (1.1, 3), (2.3, 13))
-  def test_user_privacy_epsilon_delta_consistency(self, z, k):
+  def test_user_privacy_epsilon_delta_consistency(
+      self, noise_multiplier, max_examples_per_user
+  ):
     """Tests example/user epsilons consistent with Vadhan (2017) Lemma 2.2."""
     num_epochs = 5
-    user_delta = 1e-6
+    example_delta = 1e-8
     q = 2e-4
-    user_eps = _user_privacy(
-        num_epochs,
-        noise_multiplier=z,
-        user_delta=user_delta,
-        max_examples_per_user=k,
-        poisson_subsampling_probability=q,
-    )
     example_eps = _example_privacy(
         num_epochs,
-        noise_multiplier=z,
+        noise_multiplier=noise_multiplier,
-        example_delta=user_delta / (k * math.exp(user_eps)),
+        example_delta=example_delta,
         poisson_subsampling_probability=q,
     )
-    self.assertAlmostEqual(user_eps, example_eps * k)
+
+    user_delta = math.exp(
+        math.log(example_delta)
+        + compute_dp_sgd_privacy_lib._logexpm1(
+            max_examples_per_user * example_eps
+        )
+        - compute_dp_sgd_privacy_lib._logexpm1(example_eps)
+    )
+    user_eps = _user_privacy(
+        num_epochs,
+        noise_multiplier=noise_multiplier,
+        user_delta=user_delta,
+        max_examples_per_user=max_examples_per_user,
+        poisson_subsampling_probability=q,
+    )
+    self.assertAlmostEqual(user_eps, example_eps * max_examples_per_user)
 
   def test_dp_sgd_privacy_statement_no_user_dp(self):
     statement = compute_dp_sgd_privacy_lib.compute_dp_sgd_privacy_statement(
@@ -171,7 +181,7 @@ RDP accounting:
     Epsilon with each example occurring once per epoch:        13.376
     Epsilon assuming Poisson sampling (*):                      1.616
 
-No user-level privacy guarantee is possible witout a bound on the number of
+No user-level privacy guarantee is possible without a bound on the number of
 examples per user.
 
 (*) Poisson sampling is not usually done in training pipelines, but assuming
@@ -201,8 +211,8 @@ RDP accounting:
 
 User-level DP with add-or-remove-one adjacency at delta = 1e-06 computed using
 RDP accounting and group privacy:
-    Epsilon with each example occurring once per epoch:       113.899
+    Epsilon with each example occurring once per epoch:        85.940
-    Epsilon assuming Poisson sampling (*):                      8.129
+    Epsilon assuming Poisson sampling (*):                      6.425
 
 (*) Poisson sampling is not usually done in training pipelines, but assuming
 that the data was randomly shuffled, it is believed the actual epsilon should be
@@ -214,11 +224,11 @@ order.
   def test_dp_sgd_privacy_statement_user_dp_infinite(self):
     statement = compute_dp_sgd_privacy_lib.compute_dp_sgd_privacy_statement(
         **DP_SGD_STATEMENT_KWARGS,
-        max_examples_per_user=9,
+        max_examples_per_user=10,
     )
     expected_statement = """\
 DP-SGD performed over 10000 examples with 64 examples per iteration, noise
-multiplier 2.0 for 5.0 epochs with microbatching, and at most 9 examples per
+multiplier 2.0 for 5.0 epochs with microbatching, and at most 10 examples per
 user.
 
 This privacy guarantee protects the release of all model checkpoints in addition