Revealed Preference

Stochastic Gradient Descent (SGD) is an optimization algorithm commonly used in machine learning and deep learning to minimize a loss function. Unlike the traditional gradient descent, which computes the gradient using the entire dataset, SGD updates the model weights using only a single sample (or a small batch) at each iteration. This makes it faster and allows it to escape local minima more effectively. The update rule for SGD can be expressed as:

where $\theta$ represents the parameters, $\eta$ is the learning rate, and $\nabla J(\theta; x^{(i)}, y^{(i)})$ is the gradient of the loss function with respect to a single training example $(x^{(i)}, y^{(i)})$. While SGD can converge more quickly than standard gradient descent, it may exhibit more fluctuation in the loss function due to its reliance on individual samples. To mitigate this, techniques such as momentum, learning rate decay, and mini-batch gradient descent are often employed.

The Jordan Normal Form (JNF) is a canonical form for a square matrix that simplifies the analysis of linear transformations. To compute the JNF of a matrix $A$, one must first determine its eigenvalues by solving the characteristic polynomial $\det(A - \lambda I) = 0$, where $I$ is the identity matrix and $\lambda$ represents the eigenvalues. For each eigenvalue, the next step involves finding the corresponding Jordan chains by examining the null spaces of $(A - \lambda I)^k$ for increasing values of $k$ until the null space stabilizes.

These chains help to organize the matrix into Jordan blocks, which are upper triangular matrices structured around the eigenvalues. Each block corresponds to an eigenvalue and its geometric multiplicity, while the size and number of blocks reflect the algebraic multiplicity and the number of generalized eigenvectors. The final Jordan Normal Form represents the matrix $A$ as a block diagonal matrix, facilitating easier computation of functions of the matrix, such as exponentials or powers.

The Riemann Mapping Theorem states that any simply connected, open subset of the complex plane (which is not all of the complex plane) can be conformally mapped to the open unit disk. This means there exists a bijective holomorphic function $f$ that transforms the simply connected domain $D$ into the unit disk $\mathbb{D}$, such that $f: D \to \mathbb{D}$ and $f$ has a continuous extension to the boundary of $D$.

More formally, if $D$ is a simply connected domain in $\mathbb{C}$, then there exists a conformal mapping $f$ such that:

This theorem is significant in complex analysis as it not only demonstrates the power of conformal mappings but also emphasizes the uniformity of complex structures. The theorem relies on the principles of analytic continuation and the uniqueness of conformal maps, which are foundational concepts in the study of complex functions.

Behavioral economics biases refer to the systematic patterns of deviation from norm or rationality in judgment, which affect the economic decisions of individuals and institutions. These biases arise from cognitive limitations, emotional influences, and social factors that skew our perceptions and behaviors. For example, the anchoring effect causes individuals to rely too heavily on the first piece of information they encounter, which can lead to poor decision-making. Other common biases include loss aversion, where the pain of losing is felt more intensely than the pleasure of gaining, and overconfidence, where individuals overestimate their knowledge or abilities. Understanding these biases is crucial for designing better economic models and policies, as they highlight the often irrational nature of human behavior in economic contexts.

Batch Normalization is a technique used to improve the training of deep neural networks by normalizing the inputs of each layer. This process helps mitigate the problem of internal covariate shift, where the distribution of inputs to a layer changes during training, leading to slower convergence. In essence, Batch Normalization standardizes the input for each mini-batch by subtracting the batch mean and dividing by the batch standard deviation, which can be represented mathematically as:

where $\mu$ is the mean and $\sigma$ is the standard deviation of the mini-batch. After normalization, the output is scaled and shifted using learnable parameters $\gamma$ and $\beta$:

This allows the model to retain the ability to learn complex representations while maintaining stable distributions throughout the network. Overall, Batch Normalization leads to faster training times, improved accuracy, and may reduce the need for careful weight initialization and regularization techniques.

The Neoclassical Synthesis is an economic theory that combines elements of both classical and Keynesian economics. It emerged in the mid-20th century, asserting that the economy is best understood through the interaction of supply and demand, as proposed by neoclassical economists, while also recognizing the importance of aggregate demand in influencing output and employment, as emphasized by Keynesian economics. This synthesis posits that in the long run, the economy tends to return to full employment, but in the short run, prices and wages may be sticky, leading to periods of unemployment or underutilization of resources.

Key aspects of the Neoclassical Synthesis include:

• Equilibrium: The economy is generally in equilibrium, where supply equals demand.
• Role of Government: Government intervention is necessary to manage economic fluctuations and maintain stability.
• Market Efficiency: Markets are efficient in allocating resources, but imperfections can arise, necessitating policy responses.

Overall, the Neoclassical Synthesis seeks to provide a more comprehensive framework for understanding economic dynamics by bridging the gap between classical and Keynesian thought.