What’s wrong with performing our best on average?

There are many ways to set up optimal control problems when the data are subject to random phenomena. Aside from the —least robust— strategy of replacing any random quantities by their expectations, the traditional approach in stochastic programming is to view the loss or objective as a random quantity itself and then take its expectation. When an optimization problem relies strictly on the expected value of the objective, the resulting control signal will likely succeed for the bulk of the ensemble. However, while such a control signal would perform well upon repeated application —assuming the data distribution does not change— it permits arbitrarily large failures in scenarios arising from the "tails" of the parameter distribution.

In precision engineering and safety-critical AI, this is a significant vulnerability. If a broadcast pulse in a quantum computer is optimized merely for the "average" qubit, it leaves the system prone to high error rates on physical outliers. Achieving true reliability requires us to bridge the gap between optimizing for the mean and guaranteeing strict, uniform performance across the board.

A risk-averse mathematical framework

To address these issues, mathematicians use a wide array of tools—known as risk measures—to express risk aversion and account for outlier events and tail behavior. These ideas have been around for decades in computational management science, especially in financial planning and logistics. However, much less has been published regarding optimal control problems, despite the growing number of interesting applications.

Some of these risk measures have perhaps familiar names: standard deviation, semi-deviation, or value-at-risk. In our work, we make use of the Average Value-at-Risk (AVaR), also known as expected shortfall, tail value-at-risk, superquantile, and conditional value-at-risk.

Metaphorically, the confidence level parameter in AVaR—a real number between 0 and 1—acts as a tunable dial. By adjusting the confidence level, we can systematically interpolate between standard average performance and strict worst-case bounds. By evaluating the control action against the highest-impact outlier scenarios, the risk-averse framework provides robust stability without becoming so conservative that it ignores favorable outcomes.

Transitioning from risk-neutral to risk-averse optimization introduces significant mathematical complexities, particularly due to the "non-smooth" nature of these risk objectives. A major contribution of our paper is providing the rigorous topological proofs required for control-affine systems. By formally characterizing the control-to-state mapping, we establish the analytical foundation needed to guarantee that modern, infinite-dimensional optimization algorithms can actually converge on these robust solutions.