Convex optimization multiplicative weights

  • What is the multiplicative weight approach?

    The method assigns initial weights to the experts (usually identical initial weights), and updates these weights multiplicatively and iteratively according to the feedback of how well an expert performed: reducing it in case of poor performance, and increasing it otherwise..

  • This algorithm differs from the Weighted Average Algorithm by the learning rate for updating weights.
    The weights are updated by the rule: wk:=wkβλk, where β=11+√2lnK/L, and then they are normalized.
Oct 23, 2019Online convex optimization deals with the following setup: we want to design an algorithm that, at each discrete time step t = 1,2, , 
The multiplicative weights method is usually used to solve a constrained optimization Hannan's algorithm and multiplicative weights. Online convex  History and backgroundGeneral setupAlgorithm analysisApplications
The multiplicative weights update method is an algorithmic technique most commonly used for decision making and prediction, and also widely deployed in game  History and backgroundGeneral setupAlgorithm analysisApplications

How can a multiplicative weights algorithm solve a constrained optimization problem?

The main idea is to apply the Multiplicative Weights framework in a “nested” fashion: one can solve a constrained optimization problem by invoking the MW algorithm on an subset of constraints (the “outer” constraints) in the manner of Section 3

3, where the domain is now defined by the rest of the constraints (the “inner” constraints)

What is Online convex optimization?

We study online convex optimization where the possible actions are trace-one elements in a symmetric cone, generalizing the extensively-studied experts setup and its quantum counterpart

What is the multiplicative weights update method?

The Multiplicative Weights Update Method: a Meta Algorithm and Applications Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights

Their analysis are usually very similar and rely on an exponential potential function

Algorithmic technique


The multiplicative weights update method is an algorithmic technique most commonly used for decision making and prediction, and also widely deployed in game theory and algorithm design.
The simplest use case is the problem of prediction from expert advice, in which a decision maker needs to iteratively decide on an expert whose advice to follow.
The method assigns initial weights to the experts, and updates these weights multiplicatively and iteratively according to the feedback of how well an expert performed: reducing it in case of poor performance, and increasing it otherwise.
It was discovered repeatedly in very diverse fields such as machine learning, optimization, theoretical computer science, and game theory.

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