diff --git a/report/figs/f/ravine_graph.svg b/report/figs/f/ravine_graph.svg
new file mode 100644
index 0000000000000000000000000000000000000000..97ecb458b81f9207fa84caa3eadea5cb7ab8c119
--- /dev/null
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diff --git a/report/report.qmd b/report/report.qmd
index b3b32c9d19a163267f3ef361a487348576f9094b..13714a412d4146d6d464f5557b4e18bea54bc8c5 100644
--- a/report/report.qmd
+++ b/report/report.qmd
@@ -297,22 +297,45 @@ que des minimaux locaux.
 
 Le problèmes des ravines auxquelles nous avons fait référence précédemment est
 illustré à l'aide la @fig-base-high-lr. Suite au fait que la fonction 
-$f(x, y) = x^2 + 5y^2$ présente une non-isotropie (en d'autres termes le 
+$f(x, y) = x^2 + 5y^2$ présente une non-isotropie (en d'autres termes, le 
 gradient n'est pas uniforme dans toutes les dimensions), nous pouvons voir que
 l'algorithme de descente simple avec un taux d'apprentissage de $\lambda = 0.1$
-arrive très rapidement à descendre dans la "vallée", au bout d'une **seule
+arrive très rapidement à descendre dans la "vallée", au bout d'**une seule
 itération** suite au fait que le gradient est très prononcé dans la direction
 de l'axe $y$. Cependant pour pouvoir par la suite converger vers le minimum,
-72 itérations supplémentaires sont nécessaires car le gradient dans la direction
-de l'axe $x$ est comparativement très faible. Ceci implique donc le fait que
-la convergence n'est pas forcément garantie dépendamment de la fonction sur
-laquelle la descente est effectuée.
+**72** itérations supplémentaires sont nécessaires car le gradient dans la direction
+de l'axe $x$ est comparativement très faible. 
+
+Une seconde problématique est aussi soulevée. Si la nature non-isotropique de
+la fonction est couplée à un taux d'apprentissage "trop" élevée, ceci peut
+potentiellement mener à une divergence de la descente comme l'illustre les
+graphiques ci-dessous.
+
+::: {.callout-important}
+Il est pertinent de noter le fait que le taux d'apprentissage $\lambda$ ne fut
+augmenté que d'un dixième par rapport à la @fig-base-high-lr. Cet ajustement
+léger mène donc à une divergence.
+:::
+
+:::: {#fig-ravine layout="[1,1]"}
+
+![Graphique de la fonction](./figs/f/ravine_graph.svg){#fig-ravine-graph width="70%"}
+
+![Lignes de niveaux de la fonction](./figs/f/ravine_topology.svg){#fig-ravine-topology width="70%"}
+
+Cas de divergence de la descente
+::::
 
 ## Plateaux
 
 Concernant la problématique des plateaux, celle-ci est assez explicite. Si
-$||\nabla{f}|| \approx 0$, alors les diverses méthodes auront beaucoup de
-peine à avancer / itérer. Ceci est très bien illustré à l'aide de l'@eq-func-well.
+$\nabla{f} \approx \vec{0}$, alors les diverses méthodes auront beaucoup de
+peine à avancer / itérer. Ce phénomène peut être illustré à l'aide de la
+fonction $f(x, y) = 1 - \exp(-10x^2 - y^2)$ dont la particularité est qu'elle
+est extrêmement plate sauf dans le voisinage de $(0, 0)$ où se situe son 
+minimum global. Par la suite nous effectuerons tout de même une descente réussie
+sur cette fonction cependant le point de départ se situera plus près de la zone
+de l'extremum.
 
 <!-- $$ -->
 <!-- \nabla{f}(2, 2) &= \begin{bmatrix} 20e^{-10x^{2} - y^{2}}x\\ 2e^{-10x^{2} - y^{2}}y \end{bmatrix} \\ -->
@@ -328,33 +351,34 @@ peine à avancer / itérer. Ceci est très bien illustré à l'aide de l'@eq-fun
 Illustration du piège posé par un plateau
 ::::
 
-La @fig-flatness-issue montrent bien le fait qu'en partant du point $(2, 2)$
+Les ci-dessus illustrenet bien le fait qu'en partant du point $(2, 2)$
 de la fonction, même après $1 \cdot 10^{5} - 1$ itérations, les quatres méthodes
-n'ont su avancé dans la direction du minimum suite au fait que la norme du
-gradient dans cette région de la fonction est extrêmement faible.
-
-\newpage
+n'ont su avancé dans la direction du minimum suite au fait que le gradient
+présente peu de variation et par-dessus tout, le fait qu'il est presque égale
+au vecteur nul.
 
 ## Minimums locaux
 
-Lorem ipsum dolor sit amet, officia excepteur ex fugiat reprehenderit enim 
-labore culpa sint ad nisi Lorem pariatur mollit ex esse exercitation amet. 
-Nisi anim cupidatat excepteur officia. Reprehenderit nostrud nostrud ipsum 
-Lorem est aliquip amet voluptate voluptate dolor minim nulla est proident. 
-Nostrud officia pariatur ut officia. Sit irure elit esse ea nulla sunt ex 
-occaecat reprehenderit commodo officia dolor Lorem duis laboris cupidatat 
-officia voluptate. Culpa proident adipisicing id nulla nisi laboris ex in 
-Lorem sunt duis officia eiusmod. Aliqua reprehenderit commodo ex non excepteur 
-duis sunt velit enim. Voluptate laboris sint cupidatat ullamco ut ea 
-consectetur et est culpa et culpa duis.
+La problématique des minimums locaux est aussi assez explicite. Il se peut
+qu'une fonction en possède une multitude. Par conséquent, dépendamment du point
+de départ initial, il se peut qu'en effectuant une descente simple ou une
+descente Momentum qui n'a su accumuler assez d'inertie, on se retrouve coincé
+dans un minimum local. Ceci n'est bien évidemment pas idéal car à présent le
+but final d'atteindre un minimum global dépend aussi du point de départ choisi,
+or il s'avère que le choix de celui-ci est souvent aléatoire.
+
+Cette problématique sera illustrée à l'aide de divers graphiques lorsque nous
+effectuerons les descentes de gradient sur la fonction d'**Ackley**. La
+particularité de cette fonction est le fait qu'elle possède justement un très
+grand nombre de minimum locaux et un minimum global bien défini en $(0, 0)$.
 
 # Descentes sur diverses fonctions
 
 Après avoir pu discuter de manière détaillée des diverses méthodes de descente
-ainsi que de leurs implications sur la vitesse de convergence et les
-trajectoires tracées, nous pouvons à présent un peu nous amuser en essayant de
-trouver les minimums de fonctions non-triviales à l'aide des méthodes de
-descente étudiées.
+ainsi que des problématiques qui peuvent survenir suite aux particularités
+topologiques de la fonction à optimiser, nous pouvons à présent un peu nous
+amuser en essayant de trouver les minimums globaux de certaines fonctions
+non-triviales à l'aide des méthodes étudiées précédemment.
 
 ## Le "puit"
 
@@ -364,25 +388,33 @@ $$
 f(x, y) = 1 - \exp(-10x^2 - y^2)
 $$ {#eq-func-well}
 
+Nous avons déjà brièvement introduit cette fonction lors de la discussion de
+la problématique des plateaux. Les graphiques ci-dessous permettent de mieux
+visualiser sa particularité principale, celle d'un minimum global très prononcé
+en $(0, 0)$ mais d'une topologie extrêmement plate dans ses environs. La
+@fig-well-topology démontre cet aspect de topologie plate à travers le fait
+qu'aucune ligne de niveau ne s'affiche dans le voisinage très proche de
+l'extremum.
+
 :::: {#fig-well layout="[1,1]"}
 
 ![Graphique de la fonction "puit"](./figs/puit/graph.svg){#fig-well-graph}
 
-![Méthodes de descentes](./figs/puit/topology.svg){#fig-well-topology}
+![Lignes de niveaux de descentes](./figs/puit/topology.svg){#fig-well-topology}
 
-Descentes sur la fonction "puit" -- $f(x, y) = 1 - \exp(-10x^2 - y^2)$
+Descentes sur la fonction "puit"
 ::::
 
 ### Valeur de la fonction de coût
 
 Le graphique ci-dessous permet d'illustrer clairement l'évolution de la valeur
 de la fonction de coût $f$ par rapport au nombre d'itérations effectuées par
-chaque méthode. Nous pouvons voir qu'**Adam** ait atteint le minimum en premier,
+chaque méthode. Nous pouvons voir qu'**Adam** a atteint le minimum en premier,
 au bout de seulement $275$ itérations. **Momentum** et **Nesterov** sont tous deux
 très similaires, $\sim 2'000$ itérations. Puis finalement la méthode de base qui
-nécessité $17'452$ itérations pour y parvenir.
+nécessite $17'452$ itérations pour y parvenir.
 
-![Évolution de la valeur de la fonction de coût](./figs/puit/cost_function.svg){#fig-well-cost-function width="80%"}
+![Évolution de la valeur de la fonction de coût "puit"](./figs/puit/cost_function.svg){#fig-well-cost-function width="55%"}
 
 ## Rosenbrock
 
@@ -392,11 +424,11 @@ $$ {#eq-func-rosenbrock}
 
 :::: {#fig-rosenbrock layout="[1,1]"}
 
-![Graphique de la fonction Rosenbrock](./figs/rosenbrock/graph.svg){#fig-rosenbrock-graph width="80%"}
+![Graphique de la fonction Rosenbrock](./figs/rosenbrock/graph.svg){#fig-rosenbrock-graph}
 
-![Méthodes de descentes](./figs/rosenbrock/topology.svg){#fig-rosenbrock-topology width="80%"}
+![Méthodes de descentes](./figs/rosenbrock/topology.svg){#fig-rosenbrock-topology}
 
-Descentes sur la fonction Rosenbrock
+Graphique des descentes sur la fonction de Rosenbrock
 ::::
 
 ## Ackley