Course material & Lecture notes
■ Nonlinear Optimization (MIT 6.7220 / 15.084, Spring 2024)
Introduction to the fundamentals of nonlinear optimization theory and algorithms. When applicable, emphasis is put on modern applications, especially within machine learning and its subbranches, including online learning, computational decisionmaking, and nonconvex applications in deep learning.Course material
20240206
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[L01]
Introduction
Overview of the course material; goals and applications; general form of an optimization problem; Weierstrass's theorem; computational considerations.

20240208
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[L02]
Firstorder optimality conditions
Firstorder necessary optimality conditions; the unconstrained case; halfspace constraints; a ﬁrst taste of Lagrange multipliers and duality.

20240213
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[L03]
The special case of convex functions
Deﬁnition of convexity for functions; local optimality implies global optimality; suﬃciency of ﬁrstorder optimality conditions; how to recognize a convex function.

20240215
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[L04A]
Feasibility, optimization, and separation (Part A)
Feasibility as minimization of distance from the feasible set; the computational equivalence between separation and optimization: the ellipsoid algorithm.

20240222
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[L04B]
Feasibility, optimization, and separation (Part B)
Primer on computational complexity; linear programming as a problem in NP intersection coNP; certiﬁcate of infeasibility (Farkas lemma).

20240227
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[L05]
Lagrange multipliers and KKT conditions
Optimization problems with functional constraints; Lagrangian function and Lagrange multipliers; constraint qualiﬁcations (linear independence of constraint gradients, Slater's condition).

20240229
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[L06]
Conic optimization
Conic programs and notable cases: linear programs, second order cone programs, semideﬁnite programs; selected applications.

20240305
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[L07]
Gradient descent and descent lemmas
Gradient descent algorithm; smoothness and the gradient descent lemma; computation of a point with small gradient in general (nonconvex) functions; descent in function value for convex functions: the Euclidean mirror descent lemma.

20240307
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[L08]
Acceleration and momentum
The idea of momentum; Nesterov's accelerated gradient descent; AllenZhu and Orecchia's accelerated gradient descent; practical considerations.

20240312
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[L09A]
Projected gradient descent and mirror descent (Part A)
The projected gradient descent (PGD) algorithm; distancegenerating functions and Bregman divergences; proximal steps and their properties; the mirror descent algorithm; descent lemmas for mirror descent.

20240314
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[L09B]
Projected gradient descent and mirror descent (Part B)
The projected gradient descent (PGD) algorithm; distancegenerating functions and Bregman divergences; proximal steps and their properties; the mirror descent algorithm; descent lemmas for mirror descent.

20240402
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[L10]
Stochastic gradient descent
Practical importance of stochastic gradient descent with momentum in tranining deep learning models; empirical risk minimization problems; minibatches; stochastic gradient descent lemmas; the eﬀect of variance.

20240404
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[L11]
Distributed optimization and ADMM
The setting of distributed optimization; the alternating direction method of multipliers (ADMM) algorithm; convergence analysis of ADMM in the case of convex functions.

20240409
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[L12]
Hessians, preconditioning, and Newton's method
Local quadratic approximation of objectives; Newton's method; local quadratic convergence rate; global convergence for certain classes of functions; practical considerations.

20240411
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[L13]
Adaptive preconditioning: AdaGrad and ADAM
Adaptive construction of diagonal preconditioning matrices; the AdaGrad algorithm; the convergence rate of AdaGrad; proof sketch; AdaGrad with momentum: the popular ADAM algorithm.

■ Topics in Multiagent Learning (MIT 6.S890, Fall 2023)
This new graduate course, codeveloped with Costis Daskalakis, presents the foundations of multiagent systems from a combined gametheoretic, optimization and learningtheoretic perspective, building from matrix games (such as rockpaperscissors) to stochastic games, imperfect information games, and games with nonconcave utilities. We present manifestations of these models in machine learning applications, from solving Go to multiagent reinforcement learning, adversarial learning and broader multiagent deep learning applications. We discuss aspects of equilibrium computation and learning as well as the computational complexity of equilibria. We also discuss how the different models and methods have allowed several recent breakthroughs in AI, including human and superhumanlevel agents for established games such as Go, Poker, Diplomacy, and Stratego.Course material
20230919
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[L04]
Learning in games: Algorithms
Regret matching, regret matching plus, FTRL and multiplicative weights update.

20230921
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[L05]
Learning in games: Foundations
Regret and hindsight rationality. Phiregret minimization and special cases. Connections with equilibrium computation and saddlepoint optimization.

20231024
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[L12]
Foundations of imperfectinformation extensiveform games
Complete versus imperfect information. Kuhn's theorem. Normalform and sequenceform strategies. Similarities and differences with normalform games.

20231026
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[L13]
Linear programming for Nash equilibrium in twoplayer zerosum extensiveform games
Formulation and implementation details.

20231031
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[L14]
Learning in imperfectinformation extensiveform games (Part I)
Construction of learning algorithms for extensiveform games.

20231102
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[L15]
Learning in imperfectinformation extensiveform games (Part II) and sequential irrationality
Proof of Counterfactual Regret Minimization (CFR). Introduction to sequential irrationality.

20231107
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[L16]
Equilibrium refinements and team coordination
Extensiveform perfect equilibria and quasiperfect equilibrium.

20231109
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Course Homepage
■ Computational Game Solving (CMU 15888, Fall 2021)
This new graduate course, codeveloped with Tuomas Sandholm at CMU, focuses on multistep imperfectinformation games. Imperfectinformation games are significantly more complex than perfectinformation games like chess and Go, and see emergence of signaling and deception at equilibrium. There has been tremendous progress in the AI community on solving such games since around 2003. The course covers the fundamentals and the state of the art of solving such games.Course material
20210909
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[L02]
Representation of strategies in treeform decision spaces
Behavioral and sequenceform representation of a strategy. Computation of expected utilities given the sequenceform representation (multilinearity of expected utilities). Kuhn's theorem: relationship between normalform and sequenceform strategies. Bottomup decomposition of the sequenceform polytope.

20210914
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[L03]
Hindsight rationality and regret minimization
Phiregret minimization. Special cases: external regret minimization, internal regret minimization, swap regret. Solution to convexconcave saddlepoint problems via regret minimization. Applications to bilinear saddlepoint problems such as Nash equilibrium, optimal correlation, etc.

20210916
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[L04]
Blackwell approachability and regret minimization on simplex domains
Blackwell game approach and construction of regret matching (RM), RM+.

20210921
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[L05]
Regret circuits and the Counterfactual Regret Minimization (CFR) paradigm
Treeplex case: regret circuits for Cartesian products and for convex hull. Construction of CFR and pseudocode; proof of correctness and convergence speed.

20210928
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[L07]
Optimistic/predictive regret minimization via online optimization
Online projected gradient descent. Distancegenerating functions. Predictive followtheregularizedleader (FTRL), predictive online mirror descent (OMD), and RVU bounds. Notable instantiations, e.g., optimistic hedge / multiplicative weights update. Accelerated convergence to bilinear saddle points. Dilatable global entropy.

20210930
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[L08]
Predictive Blackwell approachability
Blackwell approachability on conic domains. Using regret minimization to solve a Blackwell approachability game. Abernethy et al.’s construction. Predictive Blackwell approachability.

20211005
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[L09]
Predictive regret matching and predictive regret matching plus
Connections between followtheregularizedleader / online mirror descent and regret matching / regret matching plus. Construction of predictive regret matching and predictive regret matching plus.

20211007
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[L10]
MonteCarlo CFR and offline optimization techniques
Regret minimization with unbiased estimators of the utilities. Gametheoretic utility estimators (external sampling, outcome sampling). Offline optimization methods for twoplayer zerosum games. Accelerated firstorder saddlepoint solvers (excessive gap technique, mirror prox). Linear programming formulation of Nash equilibrium strategies. Payoff matrix sparsification technique.

20211111
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[L19]
Sequential irrationality and Nash equilibrium refinements
Sequential irrationality. Tremblinghand equilibrium refinements: quasiperfect equilibrium (QPE) and extensiveform perfect equilibrium (EFPE). Relationships among refinements. Computational complexity. Tremblinghand linear program formulation of QPE and EFPE. Scalable exact algorithms for QPE, and EFPE.

20211116
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[L20]
Correlated strategies and team coordination
Team maxmin equilibrium and TMECor; why the latter is often significantly better. Realization polytope: low dimensional but only the vertices are known and not the constraints.

Course Homepage
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