# Minimizing Regret ##TOP##

I decided to take the Regret Minimization Framework one step further and think about how my 80-year-old self would look back on the decisions I made with my time, and how I can alter my habits and activities to minimize future regret.

## Minimizing regret

So what will your last words be? And who will be in the room to hear them? Remember how short and fragile life is. As difficult as this is to think about and say, some of you reading this email right now will not be here next year. Make every moment count with your family. Take a moment now to ponder 10 ways to minimize your regrets at the end of your life.

A wise man has the ability to visualize outcomes before they occur. Never make decisions from a self-centered perspective. Self-centered decisions often will grant you short-term gain but create long-term damage for you and everyone around you. Always hedge your bets toward the wise thing to do. You will have fewer regrets.

An important problem in sequential decision-making under uncertainty is to use limited data to compute a safe policy, i.e., a policy that is guaranteed to perform at least as well as a given baseline strategy. In this paper, we develop and analyze a new model-based approach to compute a safe policy when we have access to an inaccurate dynamics model of the system with known accuracy guarantees. Our proposed robust method uses this (inaccurate) model to directly minimize the (negative) regret w.r.t. the baseline policy. Contrary to the existing approaches, minimizing the regret allows one to improve the baseline policy in states with accurate dynamics and seamlessly fall back to the baseline policy, otherwise. We show that our formulation is NP-hard and propose an approximate algorithm. Our empirical results on several domains show that even this relatively simple approximate algorithm can significantly outperform standard approaches.

We consider combinatorial optimization problems with uncertain parameters of the objective function, where for each uncertain parameter an interval estimate is known. It is required to find a solution that minimizes the worst-case relative regret. For minmax relative regret versions of some subset-type problems, where feasible solutions are subsets of a finite ground set and the objective function represents the total weight of elements of a feasible solution, and for the minmax relative regret version of the problem of scheduling n jobs on a single machine to minimize the total completion time, we present a number of structural, algorithmic, and complexity results. Many of the results are based on generalizing and extending ideas and approaches from absolute regret minimization to the relative regret case.

We study a general adversarial online learning problem, in which we are given a decision set X' in a reflexive Banach space X and a sequence of reward vectors in the dual space of X. At each iteration, we choose an action from X', based on the observed sequence of previous rewards. Our goal is to minimize regret, defined as the gap between the realized reward and the reward of the best fixed action in hindsight. Using results from infinite dimensional convex analysis, we generalize the method of Dual Averaging (or Follow the Regularized Leader) to our setting and obtain upper bounds on the worst-case regret that generalize many previous results. Under the assumption of uniformly continuous rewards, we obtain explicit regret bounds in a setting where the decision set is the set of probability distributions on a compact metric space S. Importantly, we make no convexity assumptions on either the set S or the reward functions. We also prove a general lower bound on the worst-case regret for any online algorithm. We then apply these results to the problem of learning in repeated two-player zero-sum games on compact metric spaces. In doing so, we first prove that if both players play a Hannan-consistent strategy, then with probability 1 the empirical distributions of play weakly converge to the set of Nash equilibria of the game. We then show that, under mild assumptions, Dual Averaging on the (infinite-dimensional) space of probability distributions indeed achieves Hannan-consistency.

The regret minimization framework is really more of a tripod that balances the business, relationship management, and where you are in your professional and personal life. Entrepreneurial life is always a bit of a high wire act, but these focal points do distill the various components into a simple formula.

While our goal is to move as fast as we can and minimize regret, if you can program machine-learning to focus on a percentage of your operations, it will freeyou up for the randomness that is the most rewarding and potentially most effective areas of marketing.