Update theorem 5 & 6 in ch6

notebooks/ch6.ipynb

@@ -288,15 +288,15 @@
     "- Then for any $\\delta' \\geq 0$, the entire $k$-fold adaptive composition satisfies $(\\epsilon', \\delta')$-differential privacy, where:\n",
     "\n",
     "\\begin{align}\n",
-    "\\epsilon' = 2\\epsilon \\sqrt{2k \\log(1/\\delta')}\n",
+    "\\epsilon' = \\epsilon \\sqrt{2k \\ln(1/\\delta')} + k \\epsilon (e^{\\epsilon} - 1)\n",
     "\\end{align}\n",
     "```\n",
     "\n",
     "Plugging in $\\epsilon = 1$ from the example above, and setting $\\delta' = 10^{-5}$, we get:\n",
     "\n",
     "\\begin{align}\n",
-    "\\epsilon' =& 2 \\sqrt{1000 \\log(100000)}\\\\\n",
-    "\\approx& 214.59\n",
+    "\\epsilon' =& \\sqrt{1000 \\ln(100000)} + 500 \\times (e - 1)\\\\\n",
+    "\\approx& 966.44\n",
     "\\end{align}\n",
     "\n",
     "So advanced composition derives a much lower bound on $\\epsilon'$ than sequential composition, *for the same mechanism*.  What does this mean? It means that the bounds given by sequential composition are *loose* - they don't tightly bound the *actual* privacy cost of the computation. In fact, advanced composition also gives loose bounds - they're just slightly *less* loose than the ones given by sequential composition.\n",
@@ -388,7 +388,7 @@
     "- Then for any $\\delta' \\geq 0$, the entire $k$-fold adaptive composition satisfies $(\\epsilon', k\\delta + \\delta')$-differential privacy, where:\n",
     "\n",
     "\\begin{align}\n",
-    "\\epsilon' = 2\\epsilon \\sqrt{2k \\log(1/\\delta')}\n",
+    "\\epsilon' = \\epsilon \\sqrt{2k \\ln(1/\\delta')} + k \\epsilon (e^{\\epsilon} - 1)\n",
     "\\end{align}\n",
     "```\n",
     "\n",