# Risk-Averse Learning with Varying Risk Levels

arXiv: https://arxiv.org/abs/2512.22986 · HTML: https://mlsanity.com/paper/2512.22986/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 29 references that are in the map.
- First posted on arXiv 28 Dec 2025.

## Builds on (most cited first, 12 of 12)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Non-stationary Stochastic Optimization](https://mlsanity.com/paper/1307.5449/) (2013)
- [An Online Convex Optimization Approach to Dynamic Network Resource Allocation](https://mlsanity.com/paper/1701.03974/) (2017)
- [Decision-Dependent Risk Minimization in Geometrically Decaying Dynamic Environments](https://mlsanity.com/paper/2204.08281/) (2022)
- [Doubly Optimal No-Regret Online Learning in Strongly Monotone Games with Bandit Feedback](https://mlsanity.com/paper/2112.02856/) (2021)
- [Risk-Averse Stochastic Convex Bandit](https://mlsanity.com/paper/1810.00737/) (2018)
- [On Optimizing the Conditional Value-at-Risk of a Maximum Cost for Risk-Averse Safety Analysis](https://mlsanity.com/paper/2106.00776/) (2021)
- [Learning of Nash Equilibria in Risk-Averse Games](https://mlsanity.com/paper/2403.10399/) (2024)
- [A Zeroth-Order Momentum Method for Risk-Averse Online Convex Games](https://mlsanity.com/paper/2209.02838/) (2022)
- [Distributionally Time-Varying Online Stochastic Optimization under Polyak-Lojasiewicz Condition with Application in Conditional Value-at-Risk Statistical Learning](https://mlsanity.com/paper/2309.09411/) (2023)
- [Risk-averse Learning with Non-Stationary Distributions](https://mlsanity.com/paper/2404.02988/) (2024)
- [Simulation Optimization of Risk Measures with Adaptive Risk Levels](https://mlsanity.com/paper/1608.00663/) (2016)

## Abstract (arXiv)

In safety-critical decision-making, the environment may evolve over time, and the learner adjusts its risk level accordingly. This work investigates risk-averse online optimization in dynamic environments with varying risk levels, employing Conditional Value-at-Risk (CVaR) as the risk measure. To capture the dynamics of the environment and risk levels, we employ the function variation metric and introduce a novel risk-level variation metric. Two information settings are considered: a first-order scenario, where the learner observes both function values and their gradients; and a zeroth-order scenario, where only function evaluations are available. For both cases, we develop risk-averse learning algorithms with a limited sampling budget and analyze their dynamic regret bounds in terms of function variation, risk-level variation, and the total number of samples. The regret analysis demonstrates the adaptability of the algorithms in non-stationary and risk-sensitive settings. Finally, numerical experiments are presented to demonstrate the efficacy of the methods.