This lecture examines learning dynamics in games. Wednesday covered best-response dynamics in potential games. Today focuses on no-regret dynamics, applicable beyond potential games, converging quickly to approximate coarse correlated equilibria. The core concept is single-player regret minimization against an adversary, aiming to minimize the difference between an algorithm's cost and the best fixed action's cost in hindsight. A multiplicative weights algorithm achieves optimal regret bounds, extending to multiplayer games where the empirical distribution of outcomes approximates a coarse correlated equilibrium, enabling application of price of anarchy bounds. This segment introduces the core problem: determining if players reach equilibrium in games and justifying equilibrium analysis. It highlights the importance of speed of convergence and introduces the concept of "price of anarchy" analysis as a special case when the objective function value is of concern. This segment introduces no-regret dynamics as an alternative to best-response dynamics, emphasizing their broader applicability beyond potential games and their rapid convergence to approximate equilibria (coarse correlated equilibria).This segment lays the groundwork by focusing on the single-player scenario of regret minimization, setting the stage for the later extension to multi-player games. It clearly defines the scope of the upcoming discussion.This segment formally defines the single-player game setup, including the action set, time horizon, and the adversary's role in selecting cost vectors. It emphasizes the importance of randomization in the player's strategy. This segment acknowledges the seemingly unfair nature of the adversarial model, where the player moves first and the adversary can penalize likely strategies. It sets the stage for exploring what can be achieved in this setting.This segment introduces the crucial concept of external regret, contrasting it with the unrealistic benchmark of comparing to the best action sequence in hindsight. It explains why this alternative benchmark is more appropriate and tractable. This segment explains the two fundamental principles behind no-regret algorithms: being sensitive to past performance and aggressively punishing poorly performing actions. The speaker highlights that while the first principle is crucial for achieving no regret, the second is essential for obtaining optimal bounds. The multiplicative weights algorithm, introduced later, is presented as a simple and natural implementation of these principles. This segment details the core mechanism of the multiplicative weights algorithm. It explains how weights are assigned to actions, representing their credibility, and how these weights are updated based on the cost incurred by each action. The algorithm's reliance on probability proportional to weights and the use of an epsilon parameter for tuning the update are clearly explained. This segment provides an intuitive explanation of the multiplicative weights algorithm, using a simplified scenario where costs are either zero or one. It illustrates how the algorithm's behavior changes based on the value of the epsilon parameter, interpolating between random exploration (epsilon close to zero) and deterministic exploitation of past successes (epsilon close to one). This segment sets the stage for the proof of the algorithm's no-regret property. It introduces the concept of a proxy function, gamma (sum of weights), which will be used to connect the algorithm's cost to the cost of the best fixed action in hindsight. The speaker emphasizes the need to relate these two seemingly different quantities. This segment presents the first step in the proof, focusing on establishing a lower bound for the sum of weights (gamma) based on the existence of a good fixed action. It demonstrates how the weight of a good action remains relatively high, providing a lower bound for the total sum of weights. This segment concludes the proof by combining the results from the previous steps to derive the regret bound. It involves using Taylor expansions to simplify the expressions and ultimately obtain a bound on the algorithm's cost relative to the optimal cost. The speaker highlights the use of Taylor approximations, a common technique in such proofs. This segment details the crucial step of setting the epsilon parameter in the multiplicative weights algorithm to balance error terms, achieving a regret bound of 2√(log n/T). The discussion connects this theoretical result to practical implications, showing how the required number of rounds to achieve a target regret scales logarithmically with the number of strategies. This segment introduces "no regret dynamics," a game-theoretic framework where each player employs a no-regret algorithm. It clarifies the synchronous decision-making process and emphasizes that the specific no-regret algorithm used by each player is not critical; only the no-regret guarantee matters. The explanation of how cost vectors are fed into the algorithm is particularly insightful. This segment details the second step of the proof, establishing a connection between the algorithm's cost and the rate of decrease in the sum of weights. It explains how a rapid decrease in the sum of weights is directly linked to the algorithm incurring high costs. This segment explains how cost vectors are determined within the no-regret dynamics framework. It details how each player, at each time step, observes the actions of other players and calculates the cost of each of its possible actions, given the other players' choices. This cost vector is then fed into the player's no-regret algorithm to determine its next action. This segment establishes a crucial connection between no-regret dynamics and coarse correlated equilibria (CCE). It shows that the empirical distribution of outcomes generated by no-regret dynamics approximates a CCE. The explanation includes a formal definition of CCE and its relationship to other equilibrium concepts, highlighting its significance as the most permissive equilibrium concept. This segment connects the theoretical results on no-regret dynamics to practical implications regarding the price of anarchy (PoA). It demonstrates that for smooth games, the PoA bounds derived for Nash equilibria also apply to the outcome sequences generated by no-regret dynamics. This result holds without the restrictive assumptions of potential games, significantly broadening the applicability of PoA bounds.