Fix missing sign in cross-entropy & NLL

notebooks/generative-vs-discriminative-circuit.ipynb

@@ -268,7 +268,7 @@
     "The discriminative training loss is the **cross-entropy** between the target distribution, $p^*$, and our predicted distribution, $p$:\n",
     "$$\n",
     "\\begin{align}\n",
-    "\\mathcal{L}_{dis}(\\theta) &= \\frac{1}{N} \\sum_{i=1}^N \\underbrace{\\sum_{y' \\in \\{0,\\ldots,9\\}} p^*(y' \\mid \\mathbf{x}^{(i)}) \\log p(y' \\mid \\mathbf{x}^{(i)})}_{\\text{cross-entropy}}\n",
+    "\\mathcal{L}_{dis}(\\theta) &= -\\frac{1}{N} \\sum_{i=1}^N \\underbrace{\\sum_{y' \\in \\{0,\\ldots,9\\}} p^*(y' \\mid \\mathbf{x}^{(i)}) \\log p(y' \\mid \\mathbf{x}^{(i)})}_{\\text{cross-entropy}}\n",
     "\\end{align}\n",
     "$$\n",
     "\n",
@@ -286,9 +286,9 @@
     "\n",
     "$$\n",
     "\\begin{align}\n",
-    "\\mathcal{L}_{dis}(\\theta) &= \\frac{1}{N} \\sum_{i=1}^N \\sum_{y' \\in \\{0,\\ldots,9\\}} p^*(y' \\mid \\mathbf{x}^{(i)}) \\log p(y' \\mid \\mathbf{x}^{(i)}) &  \\\\\n",
-    "                  &= \\frac{1}{N}\\sum_{i=1}^N \\left( 1 \\log p(y^{(i)} \\mid \\mathbf{x}^{(i)}) + \\sum_{y' \\neq y^{(i)}} 0 \\log p(y' \\mid \\mathbf{x}^{(i)}) \\right) & \\text{$p^*$ is one-hot} \\\\\n",
-    "                  &= \\frac{1}{N} \\sum_{i=1}^N \\log p(y^{(i)} \\mid \\mathbf{x}^{(i)}) & \\text{negative log-likelihood}\n",
+    "\\mathcal{L}_{dis}(\\theta) &= -\\frac{1}{N} \\sum_{i=1}^N \\sum_{y' \\in \\{0,\\ldots,9\\}} p^*(y' \\mid \\mathbf{x}^{(i)}) \\log p(y' \\mid \\mathbf{x}^{(i)}) &  \\\\\n",
+    "                  &= - \\frac{1}{N}\\sum_{i=1}^N \\left( 1 \\log p(y^{(i)} \\mid \\mathbf{x}^{(i)}) + \\sum_{y' \\neq y^{(i)}} 0 \\log p(y' \\mid \\mathbf{x}^{(i)}) \\right) & \\text{$p^*$ is one-hot} \\\\\n",
+    "                  &= - \\frac{1}{N} \\sum_{i=1}^N \\log p(y^{(i)} \\mid \\mathbf{x}^{(i)}) & \\text{negative log-likelihood}\n",
     "\\end{align}\n",
     "$$\n",
     "\n",