Risk-Averse Online Optimization with Time-Varying Distributions

Core Concepts

The core message of this paper is to design a risk-averse learning algorithm that achieves sub-linear dynamic regret in online convex optimization problems with time-varying distributions, using Conditional Value at Risk (CVaR) as the risk measure.

Abstract

The paper investigates online convex optimization in non-stationary environments, where the distribution of the random cost function changes over time. It proposes a risk-averse learning algorithm that minimizes the CVaR of the cost function.
Key highlights:

The algorithm uses a zeroth-order optimization approach to estimate the CVaR gradient, as the exact gradient is generally unavailable.
It employs a restarting procedure to enable the algorithm to adapt to the changing distributions.
The distribution variation is quantified using the Wasserstein distance metric.
The dynamic regret of the algorithm is analyzed for both convex and strongly convex cost functions, showing sub-linear bounds in terms of the distribution variation.
The number of samples used to estimate the CVaR gradient is controlled by a tuning parameter, which affects the regret bound.
Numerical experiments on dynamic pricing in a parking lot are provided to demonstrate the efficacy of the proposed algorithm.

Stats

The paper does not contain any explicit numerical data or statistics. The analysis focuses on theoretical regret bounds.

Quotes

None.

Key Insights Distilled From

Risk-averse Learning with Non-Stationary Distributions

by Siyi Wang,Zi... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.02988.pdf

Deeper Inquiries

How can the proposed algorithm be extended to handle constraints or multi-agent settings

To extend the proposed algorithm to handle constraints or multi-agent settings, we can introduce additional constraints to the optimization problem. For constraints, we can incorporate them into the cost function or introduce penalty terms for violating constraints. This can be achieved by modifying the objective function to include the constraints and updating the algorithm to ensure that the constraints are satisfied during optimization.
In the case of multi-agent settings, we can consider a game-theoretic approach where each agent's decision affects the others. The algorithm can be adapted to account for the interactions between agents, potentially using techniques from multi-agent reinforcement learning or game theory. By modeling the interactions and dependencies between agents, the algorithm can optimize decisions in a collaborative or competitive environment.

What are the limitations of using the Wasserstein distance metric to capture distribution variations, and are there alternative metrics that could be explored

While the Wasserstein distance metric is a powerful tool for quantifying the dissimilarity between probability distributions, it has some limitations. One limitation is computational complexity, especially for high-dimensional distributions. Calculating the Wasserstein distance can be computationally intensive, making it challenging for large-scale problems.
Additionally, the Wasserstein distance may not capture all aspects of distribution variations, especially in cases where distributions have different shapes or modes. Alternative metrics that could be explored include Kullback-Leibler divergence, total variation distance, or Hellinger distance. These metrics offer different perspectives on distribution differences and may provide complementary insights to the Wasserstein distance.

Can the risk-averse learning framework be applied to other types of online optimization problems beyond convex functions, such as non-convex or combinatorial problems

The risk-averse learning framework can be applied to a wide range of online optimization problems beyond convex functions. For non-convex problems, the framework can be adapted by considering different risk measures or modifying the algorithm to handle the non-convexity of the cost function. Techniques such as stochastic gradient descent or evolutionary algorithms can be used to optimize non-convex functions while incorporating risk-averse considerations.
In the case of combinatorial problems, the risk-averse learning framework can be applied by defining appropriate risk measures for the combinatorial space. This may involve considering the uncertainty in selecting combinations or permutations of elements and optimizing the decision-making process to minimize the risk of unfavorable outcomes. Techniques such as integer programming or dynamic programming can be used to address combinatorial optimization problems within a risk-averse framework.