# Convex duality for stochastic shortest path problems in known and unknown environments

arXiv: https://arxiv.org/abs/2208.00330 · HTML: https://mlsanity.com/paper/2208.00330/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 38 references that are in the map.
- First posted on arXiv 31 Jul 2022.

## Builds on (most cited first, 10 of 10)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [A Unifying View of Optimism in Episodic Reinforcement Learning](https://mlsanity.com/paper/2007.01891/) (2020)
- [Near-optimal Regret Bounds for Stochastic Shortest Path](https://mlsanity.com/paper/2002.09869/) (2020)
- [No-Regret Exploration in Goal-Oriented Reinforcement Learning](https://mlsanity.com/paper/1912.03517/) (2019)
- [Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret](https://mlsanity.com/paper/2104.11186/) (2021)
- [Minimax Regret for Stochastic Shortest Path](https://mlsanity.com/paper/2103.13056/) (2021)
- [Stochastic Shortest Path with Adversarially Changing Costs](https://mlsanity.com/paper/2006.11561/) (2020)
- [Minimax Regret for Stochastic Shortest Path with Adversarial Costs and Known Transition](https://mlsanity.com/paper/2012.04053/) (2020)
- [Implicit Finite-Horizon Approximation and Efficient Optimal Algorithms for Stochastic Shortest Path](https://mlsanity.com/paper/2106.08377/) (2021)
- [Online Learning for Stochastic Shortest Path Model via Posterior Sampling](https://mlsanity.com/paper/2106.05335/) (2021)

## Abstract (arXiv)

This paper studies Stochastic Shortest Path (SSP) problems in known and unknown environments from the perspective of convex optimisation. It first recalls results in the known parameter case, and develops understanding through different proofs. It then focuses on the unknown parameter case, where it studies extended value iteration (EVI) operators. This includes the existing operators used in Rosenberg et al. [26] and Tarbouriech et al. [31] based on the l-1 norm and supremum norm, as well as defining EVI operators corresponding to other norms and divergences, such as the KL-divergence. This paper shows in general how the EVI operators relate to convex programs, and the form of their dual, where strong duality is exhibited. This paper then focuses on whether the bounds from finite horizon research of Neu and Pike-Burke [21] can be applied to these extended value iteration operators in the SSP setting. It shows that similar bounds to [21] for these operators exist, however they lead to operators that are not in general monotone and have more complex convergence properties. In a special case we observe oscillating behaviour. This paper generates open questions on how research may progress, with several examples that require further examination.