Autoregressive Language Models

Autoregressive Language Models

• Autoregressive (AR) language models predict the next word based on previous words.

Definition

• Predicts the probability of the next word given the preceding words: $P(x_t \mid x_{t-1},..., x_1)$
• $x_t$ represents the word at time step t.

Types

• Forward Autoregressive Language Models: Predicts the next word from previous words, using input sequence $x_1, x_2,..., x_{t-1}$ to predict a probability distribution for the next word $x_t$.
• Backward Autoregressive Language Models: Predicts the previous word from subsequent words with input sequence $x_t, x_{t+1},..., x_T$ to predict a probability distribution for the previous word $x_{t-1}$.

Example

• Given the sentence "The quick brown fox", a forward model predicts "fox" from "The quick brown."
• A backward model predicts "quick" from "brown fox."

Advantages

• Models can generate fluent and coherent text.
• Models can be trained on substantial text datasets.
• Models can be applied in text generation, machine translation, and question answering.

Disadvantages

• Training can be computationally demanding.
• There is a potential for repetitive text generation.
• Models can reflect biases present in training data.

Reaction Rate

• For $aA + bB \rightarrow cC + dD$, Rate $= -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt}$

Factors Affecting Reaction Rates

• Concentration affects reaction rates; higher concentration results in faster rates.
• Temperature affects reaction rates; higher temperature results in faster rates.
• The surface area of solid reactants impacts reaction speed; larger surface areas increase rate.
• Catalysts speed up reactions.

Rate Law Expression

• Rate = $k[A]^m[B]^n$, where $k$ is the rate constant, $m$ is the order with respect to A, and $n$ is the order with respect to B; $m + n$ is the overall reaction order.

Determining Reaction Order

• The method of initial rates compares initial rates for varying initial concentrations.
• Integrated rate laws use concentration versus time data.

Integrated Rate Laws

• Order 0: Rate = $k$, Integrated Rate Law: $[A]_t = -kt + [A]_0$, Linear Plot: $[A]t$ vs. t, Slope: $-k$, Half-Life: $t{1/2} = [A]_0/2k$
• Order 1: Rate = $k[A]$, Integrated Rate Law: $ln[A]_t = -kt + ln[A]_0$, Linear Plot: $ln[A]t$ vs. t, Slope: $-k$, Half-Life: $t{1/2} = 0.693/k$
• Order 2: Rate = $k[A]^2$, Integrated Rate Law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$, Linear Plot: $\frac{1}{[A]t}$ vs. t, Slope: $k$, Half-Life: $t{1/2} = 1/k[A]_0$

Arrhenius Equation

• Relates rate constant to temperature: $k = Ae^{-E_a/RT}$, where $k$ is the rate constant, $A$ is the frequency factor, $E_a$ is the activation energy, $R$ is the gas constant (8.314 J/mol·K), and $T$ is the temperature in Kelvin.

Two-Point Form

• $ln(\frac{k_2}{k_1}) = \frac{E_a}{R} (\frac{1}{T_1} - \frac{1}{T_2})$

Elementary Steps

• Unimolecular reactions involve one molecule: $A \rightarrow products$; Rate $= k[A]$.
• Bimolecular reactions involve two molecules: $A + B \rightarrow products$; Rate $= k[A][B]$ or $2A \rightarrow products$; Rate $= k[A]^2$.
• Termolecular reactions involve three molecules (rare).

Rate-Determining Step

• The slowest reaction step controls the reaction's overall rate

Catalysis

• Homogeneous catalysts exist in the same phase as reactants.
• Heterogeneous catalysts exist in distinctly different phases from reactants.