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Classical Descriptive Set Theory

Optimal Transport: A Semi-Discrete Approach

In this chapter we describe some of the asymptotic properties of Bayes procedures. These are obtained by using on the parameter set Θ a finite positive measure μ and minimizing the average risk $\int R(\theta,\rho)\mu(d\theta)$. (See Chapter 2 for notation). The procedure ρ that achieves this minimum will of course depend on the choice of μ. However the literature contains numerous statements to the effect that, for large samples, the choice of μ matters little. This cannot be generally true, but we start with a proposition to this effect. If instead of μ one uses A dominated by μ and if the density $\frac{d\lambda}{d\mu}$ can be closely estimated, then a procedure that is nearly Bayes for μ is also nearly Bayes for λ.

https://link.springer.com/content/pdf/10.1007%2F978-1-4684-0377-0_7.pdf

On Bayes Procedures

Combining different models is a widely used paradigm in machine learning applications. While the most common approach is to form an ensemble of models and average their individual predictions, this approach is often rendered infeasible by given resource constraints in terms of memory and computation, which grow linearly with the number of models. We present a layer-wise model fusion algorithm for neural networks that utilizes optimal transport to (soft-) align neurons across the models before averaging their associated parameters. 
We show that this can successfully yield "one-shot" knowledge transfer (i.e, without requiring any retraining) between neural networks trained on heterogeneous non-i.i.d. data. In both i.i.d. and non-i.i.d. settings , we illustrate that our approach significantly outperforms vanilla averaging, as well as how it can serve as an efficient replacement for the ensemble with moderate fine-tuning, for standard convolutional networks (like VGG11), residual networks (like ResNet18), and multi-layer perceptrons on CIFAR10, CIFAR100, and MNIST. Finally, our approach also provides a principled way to combine the parameters of neural networks with different widths, and we explore its application for model compression. The code is available at the following link, this https URL.

/pdf/model-fusion-via-optimal-transport-117pgna57w.pdf

Model Fusion via Optimal Transport

This paper describes a new continuous-time principal-agent model, in which the output is a diffusion process with drift determined by the agentâ€™s unobserved effort. The risk-averse agent receives consumption continuously. An optimal contract, based on the agentâ€™s continuation value as a state variable, is computed by a new method using a differential equation. During employment the output path stochastically drives the agentâ€™s continuation value until it hits a low retirement point or a high retirement point. Unlike in related discrete-time models, one can use calculus to derive comparative statics and evaluate inefficiency

A Continuous-Time Version of the Principal-Agent

We study mean field stochastic control problems where the cost function and the state dynamics depend upon the joint distribution of the controlled state and the control process. We prove suitable ...

/pdf/extended-mean-field-control-problems-stochastic-maximum-2juayxa35r.pdf

Extended Mean Field Control Problems: Stochastic Maximum Principle and Transport Perspective

Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she is left with the possibility that Q does not provide an "exact" description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? 
If we measure proximity with the usual Wasserstein distance (say), the answer is NO. Models which are similar w.r.t. Wasserstein distance may provide dramatically different information on which to base a hedging strategy. 
Remarkably, this can be overcome by considering a suitable "adapted" version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler \cite{Pf09,PfPi12,PfPi14}. It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.

Adapted Wasserstein Distances and Stability in Mathematical Finance

Assume that an agent models a financial asset through a measure ℚ with the goal to price/hedge some derivative or optimise some expected utility. Even if the model ℚ is chosen in the most skilful and sophisticated way, the agent is left with the possibility that ℚ does not provide an exact description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of ℚ? If we measure proximity with the usual Wasserstein distance (say), the answer is No. Models which are similar with respect to the Wasserstein distance may provide dramatically different information on which to base a hedging strategy. Remarkably, this can be overcome by considering a suitable adapted version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler (SIAM J. Optim. 20:1406–1420, 2009, SIAM J. Optim. 22:1–23, 2012, Multistage Stochastic Optimization, 2014). It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.

/pdf/adapted-wasserstein-distances-and-stability-in-mathematical-1c82e5l40n.pdf

Adapted Wasserstein distances and stability in mathematical finance

The optimal weak transport problem has recently been introduced by Gozlan et al. (J Funct Anal 273(11):3327–3405, 2017). We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier–Strassen Theorem of Gozlan and Juillet (On a mixture of brenier and strassen theorems. arXiv:1808.02681, 2018) to general probability measures on $$\mathbb {R}^d$$ under minimal assumptions. A driving idea behind our proofs is to consider the set of transport plans with a new (‘adapted’) topology which seems better suited for the weak transport problem and allows to carry out arguments which are close to the proofs in the classical setup.

/pdf/existence-duality-and-cyclical-monotonicity-for-weak-4ow9axrsug.pdf

Existence, duality, and cyclical monotonicity for weak transport costs

Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $\pi_1, \pi_2, \ldots$ converges weakly to a transport plan $\pi$, then $\pi$ is also optimal (between its marginals). 
Alfonsi, Corbetta and Jourdain asked whether the same property is true for the martingale transport problem. This question seems particularly pressing since martingale transport is motivated by robust finance where data is naturally noisy. On a technical level, stability in the martingale case appears more intricate than for classical transport since optimal transport plans $\pi$ are not characterized by a `monotonicity'-property of their support. 
In this paper we give a positive answer and establish stability of the martingale transport problem. As a particular case, this recovers the stability of the left curtain coupling established by Juillet. An important auxiliary tool is an unconventional topology which takes the temporal structure of martingales into account. Our techniques also apply to the the weak transport problem introduced by Gozlan, Roberto, Samson and Tetali.

Julio Backhoff-Veraguas

Papers

Extended Mean Field Control Problems: Stochastic Maximum Principle and Transport Perspective

Adapted Wasserstein Distances and Stability in Mathematical Finance

Adapted Wasserstein distances and stability in mathematical finance

Existence, duality, and cyclical monotonicity for weak transport costs

Stability of martingale optimal transport and weak optimal transport.