Fix minor typos in lab 2

labs/lab_two.ipynb

@@ -798,7 +798,7 @@
    "id": "3b3744cf-1ba6-438c-9e53-25e823ce696f",
    "metadata": {},
    "source": [
-    "We can now sample from, and thus visualize, the *conditional* probaability path."
+    "We can now sample from, and thus visualize, the *conditional* probability path."
    ]
   },
   {
@@ -1564,7 +1564,7 @@
    "metadata": {},
    "source": [
     "We have thus far used flow matching to train a model $u_t^{\\theta}(x) \\approx u_t^{\\text{ref}}$ so that $$d X_t = u_t^{\\theta}(X_t) dt $$ approximately passes through the desired marginal probability path $p_t(x)$. Now recall from lecture that we may augment the reference marginal vector field $u_t^{\\text{ref}}(x)$ with *Langevin dynamics* to add stochasticity while preserving the marginals, viz., $$dX_t = \\left[u_t^{\\text{ref}}(x) + \\frac{1}{2}\\sigma^2 \\nabla \\log p_t(x)\\right] dt + \\sigma d W_t.$$\n",
-    "Substituting our learned approximation $u_t^{\\theta}(x) \\approx u_t^{\\text{ref}}$ therefore yields \n",
+    "Substituting our learned approximation $u_t^{\\theta}(x) \\approx u_t^{\\text{ref}}(x)$ therefore yields \n",
     "$$dX_t = \\left[u_t^{\\theta}(x) + \\frac{1}{2}\\sigma^2 \\nabla \\log p_t(x)\\right] dt + \\sigma d W_t.$$\n",
     "There's just one issue, what's the marginal score $\\nabla \\log p_t(x)$? In Question 2.3, we computed the conditional score $\\nabla \\log p_t(x|z)$ of the Gaussian probability path. In the same way that we learned an approximation $u_t^{\\theta}(x) \\approx u_t^{\\text{ref}}$, we'd like to be able to learn a similar approximation $s_t^{\\theta}(x) \\approx \\nabla \\log p_t(x)$. Recall from lecture the identity $$\\nabla \\log p_t(x) = \\mathbb{E}_{z \\sim p_t(z|x)}\\left[\\nabla \\log p_t(x|z) \\right].$$ It then immediately follows that\n",
     "$$\\nabla \\log p_t(x) = \\text{argmin}_{s_t(x)} \\,\\,\\mathbb{E}_{z \\sim p(z), x \\sim p_t(x|z)} \\left[\\lVert s_t(x) - \\nabla \\log p_t(x|z)\\rVert^2\\right].$$\n",

solutions/lab_two_complete.ipynb

@@ -806,7 +806,7 @@
    "id": "3b3744cf-1ba6-438c-9e53-25e823ce696f",
    "metadata": {},
    "source": [
-    "We can now sample from, and thus visualize, the *conditional* probaability path."
+    "We can now sample from, and thus visualize, the *conditional* probability path."
    ]
   },
   {
@@ -1578,7 +1578,7 @@
    "metadata": {},
    "source": [
     "We have thus far used flow matching to train a model $u_t^{\\theta}(x) \\approx u_t^{\\text{ref}}$ so that $$d X_t = u_t^{\\theta}(X_t) dt $$ approximately passes through the desired marginal probability path $p_t(x)$. Now recall from lecture that we may augment the reference marginal vector field $u_t^{\\text{ref}}(x)$ with *Langevin dynamics* to add stochasticity while preserving the marginals, viz., $$dX_t = \\left[u_t^{\\text{ref}}(x) + \\frac{1}{2}\\sigma^2 \\nabla \\log p_t(x)\\right] dt + \\sigma d W_t.$$\n",
-    "Substituting our learned approximation $u_t^{\\theta}(x) \\approx u_t^{\\text{ref}}$ therefore yields \n",
+    "Substituting our learned approximation $u_t^{\\theta}(x) \\approx u_t^{\\text{ref}}(x)$ therefore yields \n",
     "$$dX_t = \\left[u_t^{\\theta}(x) + \\frac{1}{2}\\sigma^2 \\nabla \\log p_t(x)\\right] dt + \\sigma d W_t.$$\n",
     "There's just one issue, what's the marginal score $\\nabla \\log p_t(x)$? In Question 2.3, we computed the conditional score $\\nabla \\log p_t(x|z)$ of the Gaussian probability path. In the same way that we learned an approximation $u_t^{\\theta}(x) \\approx u_t^{\\text{ref}}$, we'd like to be able to learn a similar approximation $s_t^{\\theta}(x) \\approx \\nabla \\log p_t(x)$. Recall from lecture the identity $$\\nabla \\log p_t(x) = \\mathbb{E}_{z \\sim p_t(z|x)}\\left[\\nabla \\log p_t(x|z) \\right].$$ It then immediately follows that\n",
     "$$\\nabla \\log p_t(x) = \\text{argmin}_{s_t(x)} \\,\\,\\mathbb{E}_{z \\sim p(z), x \\sim p_t(x|z)} \\left[\\lVert s_t(x) - \\nabla \\log p_t(x|z)\\rVert^2\\right].$$\n",