Egor Gladin

Egor Gladin, Maksim Lavrik-Karmazin, Karina Zainullina et al.

The problem of constrained Markov decision process is considered. An agent aims to maximize the expected accumulated discounted reward subject to multiple constraints on its costs (the number of constraints is relatively small). A new dual approach is proposed with the integration of two ingredients: entropy regularized policy optimizer and Vaidya's dual optimizer, both of which are critical to achieve faster convergence. The finite-time error bound of the proposed approach is provided. Despite the challenge of the nonconcave objective subject to nonconcave constraints, the proposed approach is shown to converge (with linear rate) to the global optimum. The complexity expressed in terms of the optimality gap and the constraint violation significantly improves upon the existing primal-dual approaches.

Egor Gladin, Alexey Kroshnin, Jia-Jie Zhu et al.

The LogSumExp function, also known as the free energy, plays a central role in many important optimization problems, including entropy-regularized optimal transport and distributionally robust optimization (DRO). It is also the dual to the Kullback-Leibler (KL) divergence, which is widely used in machine learning. In practice, when the number of exponential terms inside the logarithm is large or infinite, optimization becomes challenging since computing the gradient requires differentiating every term. Previous approaches that replace the full sum with a small batch introduce significant bias. We propose a novel approximation to LogSumExp that can be efficiently optimized using stochastic gradient methods. This approximation is rooted in a sound modification of the KL divergence in the dual, resulting in a new $f$-divergence called the safe KL divergence. The accuracy of the approximation is controlled by a tunable parameter and can be made arbitrarily small. Like the LogSumExp, our approximation preserves convexity. Moreover, when applied to an $L$-smooth function bounded from below, the smoothness constant of the resulting objective scales linearly with $L$. Experiments in DRO and continuous optimal transport demonstrate the advantages of our approach over state-of-the-art baselines and the effective treatment of numerical issues associated with the standard LogSumExp and KL.