Add DAS gradient tracking equilibrium

src/year2/distributed-autonomous-systems/sections/_optimization.tex

@@ -718,7 +718,7 @@
         \[ \vec{r}_i^k = \nabla l_i(\z_i^k) \]
         Then, the estimate of the average signal (i.e., gradient) is given by:
         \[
-            \vec{s}_i^{k+1} = \sum_{j \in \mathcal{N}_i} a_{ij} \vec{s}_j^k + \left( \nabla l_i(\z_i^{k+1}) - \nabla l_i(\z_i^k) \right)
+            \vec{s}_i^{k+1} = \sum_{j \in \mathcal{N}_i} a_{ij} \vec{s}_j^k + \left( \nabla l_i(\z_i^{k+1}) - \nabla l_i(\z_i^k) \right) \qquad \s_i^0 = \nabla l_i(\z_i^0)
         \]
         The update step is then performed as:
         \[ \z_i^{k+1} = \sum_{j \in \mathcal{N}_i} a_{ij} \z_j^k - \alpha \vec{s}_i^k \]
@@ -748,9 +748,161 @@
                 \,\,\land\,\,
                 \rho \Vert \z_i^{k+1} - \z^* \Vert \leq \rho^k \Vert \z_i^0 - \z^* \Vert
             \]
+
+            {
+                \indenttbox
+                \begin{remark}
+                    It can be shown that gradient tracking also works with non-convex optimization and, under the correct assumptions, converges to a stationary point. 
+                \end{remark}
+            }
+
+            \begin{proof}
+                Consider the gradient tracking algorithm written in matrix form:
+                \[
+                    \begin{aligned}
+                        \z^{k+1} &= \A \z^k - \alpha \s^k \\
+                        \s^{k+1} &= \A \s^k + (\nabla \vec{l}(\z^{k+1}) - \nabla \vec{l}(\z^k))
+                    \end{aligned}
+                \]
+                where $\nabla \vec{l}(\z^k) = \begin{bmatrix} l_1(\z^k_1) & \dots & l_N(\z^k_N) \end{bmatrix}$.
+
+                % \begin{remark}
+                %     In the vector case, the Kronecker product should be applied on $\A$.
+                % \end{remark}
+
+                \begin{description}
+                    \item[Equilibrium]
+                        We want to find the equilibrium points $(\z_\text{eq}, \s_\text{eq})$ that satisfies:
+                        \[
+                            \begin{aligned}
+                                \s_\text{eq} &= \A \s_\text{eq} + \nabla \vec{l}(\z_\text{eq}) - \nabla \vec{l}(\z_\text{eq}) &\iff& (\matr{I} - \A) \s_\text{eq} = 0 \\
+                                \z_\text{eq} &= \A\z_\text{eq} - \alpha \s_\text{eq} &\iff& (\matr{I} - \A) \z_\text{eq} = -\alpha \s_\text{eq} \\
+                            \end{aligned}
+                        \]
+                        It must be that:
+                        \begin{itemize}
+                            \item $\s_\text{eq} \in \text{ker}(\matr{I} - \A) = \{ \vec{1}\beta_1 \mid \beta_1 \in \R \}$ (as $\A$ is doubly stochastic).
+                            \item $(\matr{I} - \A) \z_\text{eq} = - \alpha \vec{1} \beta_1$. As $\vec{1} (-\alpha \beta_1) \in \text{ker}(\matr{I} - \A)$, it must be that $\beta_1 = 0$ (as the image cannot be mapped into the kernel).
+                        \end{itemize}
+                        Therefore, we end up with:
+                        \[
+                            \begin{split}
+                                \s_\text{eq} &= \vec{1}\beta_1 = 0 \\
+                                \z_\text{eq} &= \A\z_\text{eq} - \alpha 0 = \matr{1} \beta_2 \quad \text{ i.e., eigenvector of $\A$} \\
+                            \end{split}
+                        \]
+
+                        In addition, by pre-multiplying the equation of $\s$ by $\vec{1}^T$, we obtain:
+                        \[
+                            \begin{split}
+                                \vec{1}^T \s^{k+1} &= \vec{1}^T \A \s^k + \vec{1}^T \nabla \vec{l}(\z^{k+1}) - \vec{1}^T \nabla \vec{l}(\z^{k}) \\
+                                &= \vec{1}^T \s^k + \vec{1}^T \nabla \vec{l}(\z^{k+1}) - \vec{1}^T \nabla \vec{l}(\z^{k}) 
+                            \end{split}
+                        \]
+                        Which shows the following invariance condition:
+                        \[ 
+                            \begin{aligned}
+                                \vec{1}^T \s^{k+1} - \vec{1}^T \nabla \vec{l}(\z^{k+1})
+                                &= \vec{1}^T \s^k - \vec{1}^T \nabla \vec{l}(\z^{k}) \\
+                                &= \vec{1}^T \s_\text{eq} - \vec{1}^T \nabla \vec{l}(\z_\text{eq}) \\
+                                &= \vec{1}^T \s^0 - \vec{1}^T \nabla \vec{l}(\z^{0}) \\
+                            \end{aligned}
+                        \]
+                        Thus, we have that:
+                        \[
+                            \begin{split}
+                                \vec{1}^T \s_\text{eq} - \vec{1}^T \nabla \vec{l}(\z_\text{eq})
+                                &= \vec{1}^T \s^0 - \vec{1}^T \nabla \vec{l}(\z^{0}) \\
+                                \iff 0 - \vec{1}^T \nabla \vec{l}(\vec{1}\beta_2) &= 0 \\
+                            \end{split}
+                        \]
+                        Then, it must be that $\z_\text{eq} = \vec{1}\beta_2$ is an optimum with $\beta_2 = z^*$.
+                
+                    \item[Stability]
+                        % Change in coordinates to avoid having $\z^{k+1}$ in $\s^{k}$. The (non-linear) transformation is:
+                        % \[
+                        %     \begin{bmatrix}
+                        %         \z^k \\ \s^k
+                        %     \end{bmatrix}
+                        %     \mapsto
+                        %     \begin{bmatrix}
+                        %         \z^k \\ \vec{\xi}^k
+                        %     \end{bmatrix}
+                        %     =
+                        %     \begin{bmatrix}
+                        %         \z^k \\ \alpha (\nabla \vec{l}(\z^k) - \s^k)
+                        %     \end{bmatrix}
+                        % \]
+
+                        % \[
+                        %     \begin{split}
+                        %         \z^{k+1} 
+                        %         &= \A\z^k - \alpha ( \frac{1}{\alpha} \vec{\xi}^k + \nabla \vec{l}(\z^k) ) \\
+                        %         \vec{\xi}^k 
+                        %         &= \alpha \nabla \vec{l}(\z^{k+1}) - \alpha (\A \s^k + \nabla \vec{l}(\z^{k+1}) - \nabla \vec{l} (\z^k)) \\
+                        %         &= - \alpha \A (-\frac{1}{\alpha} \xi^k + \nabla \vec{l}(\z^k)) + \alpha \nabla \vec{l}(\z^k) \\
+                        %         &= \A \vec{\xi}^k - \alpha(\A - \vec{I}) \nabla \vec{l}(\z^k)
+                        %     \end{split}
+                        % \]
+
+                        % In matrix form:
+                        % \[
+                        %     \begin{bmatrix}
+                        %         \z^{k+1} \\ \vec{\xi}^{k+1} = \begin{bmatrix}
+                        %             \A & \matr{I} \\ 0 & \A
+                        %         \end{bmatrix}
+                        %         \begin{bmatrix}
+                        %             \z^k \\ \vec{\xi}^k
+                        %         \end{bmatrix}
+                        %         - alpha \begin{bmatrix}
+                        %             \matr{I} \\ \A \matr{I}
+                        %         \end{bmatrix}
+                        %         \nabla \vec{l}(\z^k)
+                        %     \end{bmatrix}
+                        % \]
+                        % The initialization is:
+                        % \[
+                        %     \begin{split}
+                        %         \z^0 \in \R^N \\
+                        %         \vec{\xi}^{0} = \alpha (\nabla \vec{l}(\z^0) - \s^0) = 0
+                        %     \end{split}
+                        % \]
+                        % The equilibrium has been shifted to:
+                        % \[
+                        %     \begin{split}
+                        %         \z_\text{eq} = \vec{1} \z^* \\
+                        %         \vec{\xi}_\text{eq} = \alpha \nabla l(\vec{1} \z^*) = \alpha \begin{bmatrix}
+                        %             \nabla l_1(\z^*) \\ \vdots \\ \nabla l_N(\z^*)
+                        %         \end{bmatrix}
+                        %     \end{split}
+                        % \]
+
+
+                        % \[
+                        %     \begin{gathered}
+                        %         \begin{bmatrix}
+                        %             \z^{k+1} \\ \vec{\xi}^{k+1} = \begin{bmatrix}
+                        %                 \A & \matr{I} \\ 0 & \A
+                        %             \end{bmatrix}
+                        %             \begin{bmatrix}
+                        %                 \z^k \\ \vec{\xi}^k
+                        %             \end{bmatrix}
+                        %             \begin{bmatrix}
+                        %                 \matr{I} \\ \A \matr{I}
+                        %             \end{bmatrix}
+                        %             \u^k
+                        %         \end{bmatrix} \\
+                        %         \vec{y}^k = \begin{bmatrix}
+                        %             \matr{I} & 0
+                        %         \end{bmatrix}
+                        %         \begin{bmatrix}
+                        %             \z^k \\ \vec{\xi}^{k}
+                        %         \end{bmatrix} \\
+                        %         -- \\
+                        %         \u^k = \nabla \vec{l}(\vec{y}^k)
+                        %     \end{gathered}
+                        % \]
+                \end{description}
+            \end{proof}
         \end{theorem}
 \end{description}
-
-\begin{remark}
-    It can be shown that gradient tracking also works with non-convex optimization and, under the correct assumptions, converges to a stationary point. 
-\end{remark}

src/year2/ethics-in-ai/module2/ethics2.tex

@@ -8,9 +8,11 @@
 \begin{document}
 
     \makenotesfront
-    \include{./sections/_gdpr.tex}
-    \include{./sections/_claudette.tex}
-    \include{./sections/_discrimination.tex}
-    \include{./sections/_autonomous_vehicles.tex}
-
+    \input{./sections/_gdpr.tex}
+    \input{./sections/_claudette.tex}
+    \input{./sections/_discrimination.tex}
+    \input{./sections/_autonomous_vehicles.tex}
+    \input{./sections/_ai_act.tex}
+    \eoc
+    
 \end{document}

src/year2/ethics-in-ai/module2/sections/_ai_act.tex

@@ -0,0 +1,343 @@
+\chapter{AI Act}
+
+
+\section{Introduction}
+
+
+\subsection{General principles}
+\marginnote{General principles}
+
+Regulate the development of AI systems based on the principles of:
+\begin{itemize}
+    \item Human agency and oversight,
+    \item Technical robustness and safety,
+    \item Privacy and data governance,
+    \item Transparency,
+    \item Diversity, non-discrimination, and fairness,
+    \item Social and environmental well-being.
+\end{itemize}
+
+
+\subsection{Definitions}
+
+\begin{description}
+    \item[AI system] \marginnote{AI system}
+        Machine-based system that is designed to operate with varying levels of autonomy and adaptability. Moreover, its output is inferred from the input data.
+
+        \begin{remark}
+            Rule-based systems are excluded.
+        \end{remark}
+
+    \item[General purpose AI] \marginnote{General purpose AI}
+        AI system that exhibits significant generality and is able to perform a wide range of tasks.
+\end{description}
+
+
+\subsection{Scope}
+
+The AI Act applies to:
+\begin{itemize}
+    \item Providers who put an AI system on the EU's market, independently of their location.
+    \item Deployers of AI systems located within the EU.
+    \item Providers and deployers in third countries if the output produced is used in the EU.
+    \item Importers and distributors of AI systems.
+    \item Product manufacturers who use AI systems in their products.
+    \item Authorized representatives of providers.
+    \item People affected by AI systems in the EU.
+\end{itemize}
+
+\begin{remark}
+    The AI Act is excluded for the following areas:
+    \begin{itemize}
+        \item Military, defense, and national security,
+        \item Scientific research and development activities,
+        \item Pre-market development and testing, if done in protected environments,
+        \item International law enforcement cooperation, if fundamental rights safeguards are in place.
+    \end{itemize}
+\end{remark}
+
+\begin{remark}
+    The AI Act is a compromise between a product safety approach (e.g., minimum safety requirements, standards, \dots) and a fundamental rights approach.
+\end{remark}
+
+\begin{remark}
+    The AI Act does not introduce new individual rights.
+\end{remark}
+
+
+
+\section{Risk regulation}
+
+\begin{description}
+    \item[Risk]
+        Combination of the probability of harm and the severity of that harm.
+\end{description}
+
+
+\subsection{Risk levels}
+
+\begin{description}
+    \item[Unacceptable-risk (article 5)] \marginnote{Unacceptable-risk}
+        Includes AI systems that are used for:
+        \begin{itemize}
+            \item Deploying harmful and manipulative subliminal techniques (i.e., beyond individual cognition),
+            \item Exploiting vulnerable groups,
+            \item Social scoring,
+            \item Real-time remote biometric identification in public spaces for law enforcement purposes (with some exceptions),
+            \item Biometric categorization of protected features,
+            \item Predicting criminal offenses solely based on profiling or personality traits,
+            \item Creating facial recognition databases by scraping the Internet or CCTV footage,
+            \item Inferring emotions in workplaces or educational institutions, unless for medical or safety reasons.
+        \end{itemize}
+
+    \item[High-risk (article 6)] \marginnote{High-risk}
+        Includes the following groups of AI systems:
+        \begin{itemize}
+            \item Those used as safety components of a product or falling under the EU health and safety legislation (e.g., toys, aviation, cars, medical devices, \dots)
+            \item Those used in the following specific areas: biometric identification, critical infrastructures, education, employment, access to essential services, law enforcement, migration, and juridical and democratic processes.
+            \item Those that performs profiling of natural persons.
+        \end{itemize}
+
+        Requirements to assess the impact of a these systems are:
+        \begin{itemize}
+            \item Determining the categories of natural persons and groups affected by the system,
+            \item Checking compliance with European and national laws,
+            \item Fundamental rights risk assessment (FRIA),
+                \begin{example}[FRAIA]
+                    Questionnaire created by the Dutch government to assess the impact of AI systems on fundamental rights.
+                \end{example}
+            \item Determining the risk of harm towards vulnerable groups and the environmental impact,
+            \item Determining a plan for risk mitigation,
+            \item Creating a governance system for human oversight, complaint handling, and redress.
+        \end{itemize}
+
+        \begin{remark}
+            These requirements are still under research.
+        \end{remark}
+
+    \item[Limited-risk (article 52)] \marginnote{Limited-risk}
+        Involves AI systems that interact with users with limited effects. It includes chatbots, emotion recognition, deep fakes, \dots
+
+        Requirements are:
+        \begin{itemize}
+            \item The user must be informed that it is interacting with an AI system,
+            \item Artificial content must be labeled as generated and contain detectable watermarks,
+            \item Employers must inform workers on whether AI is used in the workplace and the reasons.
+        \end{itemize}
+    
+    \item[Minimal-risk (article 69)] \marginnote{Minimal-risk}
+        Involves AI systems with low to no effects on the user. It includes spam filters, video games, purchase recommendation systems, \dots.
+        
+        They are required to comply with the existing regulation but are not further regulated by the AI Act.
+
+        \begin{remark}
+            Providers of these systems are nonetheless encouraged to voluntarily respect high-risk requirements.
+        \end{remark}
+
+    \item[General purpose AI requirements] \marginnote{General purpose AI requirements}
+        Specific requirements for general purpose AI are:
+        \begin{itemize}
+            \item Technical documentation must be kept for training, testing, and performance,
+            \item Key information must be shared with downstream AI system providers,
+            \item A summary of the training data must be published,
+            \item Copyright compliance must be declared,
+            \item Collaboration with regulators,
+            \item Codes of practice should be provided.
+        \end{itemize}
+
+        \begin{remark}
+            There is a subgroup of general purpose AI systems that includes those that pose a systemic risk. Additional requirements for these systems are:
+            \begin{itemize}
+                \item Additional risk mitigation,
+                \item Independent system evaluation and model registration.
+            \end{itemize}
+        \end{remark}
+
+        \begin{remark}
+            If the computing power to train a model exceeds a certain threshold, that system is presumed to be a general purpose AI that poses systemic risks.
+        \end{remark}
+\end{description}
+
+
+\subsection{Enforcement}
+
+\begin{description}
+    \item[Enforcement] \marginnote{Enforcement} 
+        National supervisory authority enforces the AI Act in each member state with the support of the European Artificial Intelligence Office.
+\end{description}
+
+
+\subsection{AI regulatory sandboxes}
+
+\begin{description}
+    \item[AI sandbox] \marginnote{AI sandbox}
+        Voluntary framework organized by member states for small to medium companies to test AI systems in controlled environments.
+\end{description}
+
+
+
+\section{AI liability}
+
+\begin{remark}
+    In the case of AI systems, liability has to account for:
+    \begin{itemize}
+        \item Black-box models,
+        \item Autonomous and unpredictable models,
+        \item Multiple actors and diffused responsibility,
+        \item Lack of a clear legal framework,
+        \item Difficulty in finding the causal chain.
+    \end{itemize}
+\end{remark}
+
+
+\subsection{Liability theories}
+
+\begin{description}
+    \item[Strict liability] \marginnote{Strict liability}
+        The producer is always responsible for their product both if it is their fault or due to negligence. The injured party only has to prove that damage occurred.
+
+    \item[Fault liability] \marginnote{Fault liability}
+        The defender has to show that someone is responsible for causing damage intentionally or negligently.
+
+    \item[Mandatory insurance] \marginnote{Mandatory insurance}
+        Enforce that the product (e.g., AI system) is covered by an insurance.
+
+    \item[Compensation funds] \marginnote{Compensation funds}
+        Economic relief for the users in case of damage or bankruptcy of the company.
+\end{description}
+
+
+\subsection{Revised Product Liability Directive}
+
+\begin{description}
+    \item[Revised Product Liability Directive] \marginnote{Revised Product Liability Directive}
+        Product Liability Directive extended to software and AI systems. It is applied in all member states (i.e., maximum harmonization) and is based on the strict liability theory.
+
+        The requirements to prove for compensation are that:
+        \begin{itemize}
+            \item The product is defective,
+            \item Damage was caused,
+            \item There is a causal link between defect and damage.
+        \end{itemize}
+
+    \item[Product] \marginnote{Product}
+        The revised Product Liability Directive extends the definition of product with:
+        \begin{itemize}
+            \item Software and its updates,
+            \item Digital manufacturing files (e.g., model for 3D printers),
+            \item Digital services.
+        \end{itemize}
+
+        \begin{remark}
+            Free and non-commercial open-source software are excluded
+        \end{remark}
+
+    \item[Liable parties (article 8)] \marginnote{Liable parties}
+        The revised Product Liability Directive extends liable entities with:
+        \begin{itemize}
+            \item Any economic operator that has substantially modified the product outside the control of the manufacturer,
+            \item Distributors of defective products,
+            \item Online platforms.
+        \end{itemize}
+    
+        \begin{remark}
+            In the case of AI systems, the producer is the provider defined in the AI Act.
+        \end{remark}
+
+    \item[Types of damage (article 6)] \marginnote{Types of damage}
+        Compensation can be provided for:
+        \begin{itemize}
+            \item Death or personal injury, including psychological health.
+            \item Damage or destruction of properties, with the exception of the product itself, other components the defective product is integrated with, and products used for professional purposes only.
+            \item Destruction or corruption of data that is not used for professional purposes.
+        \end{itemize}
+
+    \item[Defectiveness (article 7)] \marginnote{Defectiveness}
+        In the case of software, liability is applied also for defects that come out after the product has been put in the market. This includes:
+        \begin{itemize}
+            \item Software updates under the manufacturer's control,
+            \item Failure to address cybersecurity vulnerabilities,
+            \item Machine learning.
+        \end{itemize}
+
+    \item[Presumption of defectiveness and causality (article 10)] \marginnote{Presumption of defectiveness and causality}
+        Defectiveness is presumed when:
+        \begin{itemize}
+            \item The manufacturer fails to comply with the obligation to disclose information,
+            \item A product does not comply with mandatory safety requirements,
+            \item Damage is caused by an obvious product malfunction.
+        \end{itemize}
+
+        A causal link is presumed when:
+        \begin{itemize}
+            \item The damage is consistent with the type of defect,
+            \item The technical or scientific complexity makes it difficult to prove liability (e.g., as with black-box models).
+        \end{itemize}
+\end{description}
+
+\begin{remark}
+    The revised Product Liability Directive does not cover:
+    \begin{itemize}
+        \item Discrimination,
+        \item Violation of privacy (that are not already covered in the GDPR),
+        \item Use of AI for professional purposes,
+        \item Sustainability effects.
+    \end{itemize}
+\end{remark}
+
+
+\subsection{AI Liability Directive}
+
+\begin{description}
+    \item[AI Liability Directive] \marginnote{AI Liability Directive}
+        Additional protection for cases not covered in the revised Product Liability Directive. It is based on the fault liability theory.
+
+        The directive has been cancelled by the EU Commission.
+\end{description}
+
+
+\begin{example}[Case study: delivery robot accident]
+    An autonomous delivery robot that is able to navigate the pavement of pedestrian areas falls on the edge and hits a bicycle courier on the cycle lane. Both the biker and the robot sustained injuries/damage.
+    
+    \begin{descriptionlist}
+        \item[AI Act] The system falls under the high-risk level (autonomous vehicle).
+        \item[Revised Product Liability Directive] \phantom{}
+            \begin{itemize}
+                \item Liability can be sought after the company deploying the robots or the one renting them.
+                \item The defect is related to the sensors/decision-making of the robot.
+                \item Injuries are both physical and possibly psychological.
+            \end{itemize}
+    \end{descriptionlist}
+\end{example}
+
+\begin{example}[Case study: smart bank]
+    A bank stores its data in the storage service provided by another company. An update released by the bank causes the corruption and loss of financial data.
+
+    An affected customer failed to make payments leading to penalties and decrease in credit score.    
+
+    \begin{descriptionlist}
+        \item[AI Act] The system does not involve AI.
+        \item[Revised Product Liability Directive] \phantom{}
+            \begin{itemize}
+                \item Liability can be sought after the bank. 
+                \item The defect is the loss of records due to the update. 
+                \item Damages are psychological and economical.
+            \end{itemize}
+    \end{descriptionlist}
+\end{example}
+
+\begin{example}[Case study: AI friend]
+    A mental health chatbot is developed to support young users. However, a flaw in the system causes the generation of inappropriate and harmful messages.
+
+    In an affected user, this lead to depression, self-harm, declining school performance, and withdrawal from social activities.
+
+    \begin{descriptionlist}
+        \item[AI Act] The system might fall under the unacceptable-risk level (manipulation) or under the high-risk level (medical diagnosis).
+        \item[Revised Product Liability Directive] \phantom{}
+            \begin{itemize}
+                \item Liability can be sought after the company deploying the system.
+                \item The defect is the flaw of the chatbot.
+                \item Damage is psychological and physical. It also involves the loss of opportunities.
+            \end{itemize}
+    \end{descriptionlist}
+\end{example}