Bounding the Greedy Strategy in Finite-Horizon String Optimization

We consider an optimization problem where the decision variable is a string of bounded length. For some time there has been an interest in bounding the performance of the greedy strategy for this problem. Here, we provide weakened sufficient conditions for the greedy strategy to be bounded by a factor of $(1-(1-1/K)^K)$, where $K$ is the optimization horizon length. Specifically, we introduce the notions of $K$-submodularity and $K$-GO-concavity, which together are sufficient for this bound to hold. By introducing a notion of \emph{curvature} $\eta\in(0,1]$, we prove an even tighter bound with the factor $(1/\eta)(1-e^{-\eta})$. Finally, we illustrate the strength of our results by considering two example applications. We show that our results provide weaker conditions on parameter values in these applications than in previous results.Comment: This paper has been accepted by 2015 IEEE CD

Performance bounds for greedy strategies in submodular optimization problems

2018 Summer.Includes bibliographical references.To view the abstract, please see the full text of the document

Computationally bounded rationality from three perspectives: precomputation, regret tradeoffs, and lifelong learning

What does it mean for a computer program to be optimal? Many fields in optimal decision making, from game theory to Bayesian decision theory, define optimal solutions which can be computationally intractable to implement or find. This is problematic, because it means that sometimes these solutions are not physically realizable. To address this problem, bounded rationality studies what it means to behave optimally subject to constraints on processing time, memory and knowledge. This thesis contributes three new models for studying bounded rationality in different contexts. The first model considers games like chess. We suppose each player can spend some time before the game precomputing (memorizing) strong moves from an oracle, but has limited memory to remember these moves. We show how to analytically quantify how randomly optimal strategies play in equilibrium, and give polynomial- time algorithms for computing a best response and an ε-Nash equilibrium. We use the best response algorithm to empirically evaluate the chess playing program Stockfish. The second model takes place in the setting of adversarial online learning. Here, we imagine an algorithm receives new problems online, and is given a computational budget to run B problem solvers for each problem. We show how to trade off the budget B for a strengthening of the algorithm’s regret guarantee in both the full and semi-bandit feedback settings. We then show how this tradeoff implies new results for Online Submodular Function Maximization (OSFM) (Streeter and Golovin, 2008) and Linear Programming. We use these observations to derive and benchmark a new algorithm for OSFM. The third model approaches bounded rationality from the perspective of lifelong learning (Chen and Liu, 2018). Instead of modelling the final solution, lifelong learning models how a computationally bounded agent can accumulate knowledge over time and attempt to solve tractable subproblems it encounters. We develop models for incrementally accumulating and learning knowledge in a domain agnostic setting, and use these models to give an abstract framework for a lifelong reinforcement learner. The framework attempts to make a step towards making the best of analytical performance guarantees, while still being able to make use of black box techniques such as neural networks which may perform well in practice