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A recent stream of structured learning approaches has improved the practical state of the art for a range of combinatorial optimization problems with complex objectives encountered in operations research. Such approaches train policies that chain a statistical model with a surrogate combinatorial optimization oracle to map any instance of the problem to a feasible solution. The key idea is to exploit the statistical distribution over instances instead of dealing with instances separately. However learning such policies by risk minimization is challenging because the empirical risk is piecewise constant in the parameters, and few theoretical guarantees have been provided so far. In this article, we investigate methods that smooth the risk by perturbing the policy, which eases optimization and improves generalization. Our main contribution is a generalization bound that controls the perturbation bias, the statistical learning error, and the optimization error. Our analysis relies on the introduction of a uniform weak property, which captures and quantifies the interplay of the statistical model and the surrogate combinatorial optimization oracle. This property holds under mild assumptions on the statistical model, the surrogate optimization, and the instance data distribution. We illustrate the result on a range of applications such as stochastic vehicle scheduling. In particular, such policies are relevant for contextual stochastic optimization and our results cover this case., Comment: 10 pages main document, 3 pages supplement

As the problem of minimizing functionals on the Wasserstein space encompasses many applications in machine learning, different optimization algorithms on $\mathbb{R}^d$ have received their counterpart analog on the Wasserstein space. We focus here on lifting two explicit algorithms: mirror descent and preconditioned gradient descent. These algorithms have been introduced to better capture the geometry of the function to minimize and are provably convergent under appropriate (namely relative) smoothness and convexity conditions. Adapting these notions to the Wasserstein space, we prove guarantees of convergence of some Wasserstein-gradient-based discrete-time schemes for new pairings of objective functionals and regularizers. The difficulty here is to carefully select along which curves the functionals should be smooth and convex. We illustrate the advantages of adapting the geometry induced by the regularizer on ill-conditioned optimization tasks, and showcase the improvement of choosing different discrepancies and geometries in a computational biology task of aligning single-cells., Comment: Accepted as Spotlight at Conference on Neural Information Processing Systems (NeurIPS 2024)

There have been many parallel streams of research studying order isomorphisms of some specific sets $\mathcal{G}$ of functions from a set $\mathcal{X}$ to $\mathbb{R}\cup\{\pm\infty\}$, such as the sets of convex or Lipschitz functions. We provide in this article a unified abstract approach inspired by $c$-convex functions. Our results are obtained highlighting the role of inf and sup-irreducible elements of $\mathcal{G}$ and the usefulness of characterizing them, to subsequently derive the structure of order isomorphisms, and in particular of those commuting with the addition of scalars. We show that in many cases all these isomorphisms $J:\mathcal{G}\to\mathcal{G}$ are of the form $Jf=g+f\circ \phi$ for a translation $g:\mathcal{X}\to\mathbb{R}$ and a bijective reparametrization $\phi:\mathcal{X}\to \mathcal{X}$. Given a reference anti-isomorphism, this characterization then allows to recover all the other anti-isomorphisms. We apply our theory to the sets of $c$-convex functions on compact Hausdorff spaces, to the set of lower semicontinuous (convex) functions on a Hausdorff topological vector space and to 1-Lipschitz functions of complete metric spaces. The latter application is obtained using properties of the horoboundary of a metric space.

Mean-field control problems have received continuous interest over the last decade. Despite being more intricate than in classical optimal control, the linear-quadratic setting can still be tackled through Riccati equations. Remarkably, we demonstrate that another significant attribute extends to the mean-field case: the existence of an intrinsic reproducing kernel Hilbert space associated with the problem. Our findings reveal that this Hilbert space not only encompasses deterministic controlled push-forward mappings but can also represent of stochastic dynamics. Specifically, incorporating Brownian noise affects the deterministic kernel through a conditional expectation, to make the trajectories adapted. Introducing reproducing kernels allows us to rewrite the mean-field control problem as optimizing over a Hilbert space of trajectories rather than controls. This framework even accommodates nonlinear terminal costs, without resorting to adjoint processes or Pontryagin's maximum principle, further highlighting the versatility of the proposed methodology.

We propose an optimization formulation for the simultaneous estimation of a latent variable and the identification of a linear continuous-time dynamic system, given a single input-output pair. We justify this approach based on Bayesian maximum a posteriori estimators. Our scheme takes the form of a convex alternating minimization, over the trajectories and the dynamic model respectively. We prove its convergence to a local minimum which verifies a two point-boundary problem for the (latent) state variable and a tensor product expression for the optimal dynamics.

We present a new class of gradient-type optimization methods that extends vanilla gradient descent, mirror descent, Riemannian gradient descent, and natural gradient descent. Our approach involves constructing a surrogate for the objective function in a systematic manner, based on a chosen cost function. This surrogate is then minimized using an alternating minimization scheme. Using optimal transport theory we establish convergence rates based on generalized notions of smoothness and convexity. We provide local versions of these two notions when the cost satisfies a condition known as nonnegative cross-curvature. In particular our framework provides the first global rates for natural gradient descent and the standard Newton's method.

Handling an infinite number of inequality constraints in infinite-dimensional spaces occurs in many fields, from global optimization to optimal transport. These problems have been tackled individually in several previous articles through kernel Sum-Of-Squares (kSoS) approximations. We propose here a unified theorem to prove convergence guarantees for these schemes. Pointwise inequalities are turned into equalities within a class of nonnegative kSoS functions. Assuming further that the functions appearing in the problem are smooth, focusing on pointwise equality constraints enables the use of scattering inequalities to mitigate the curse of dimensionality in sampling the constraints. Our approach is illustrated in learning vector fields with side information, here the invariance of a set.

It is often said that control and estimation problems are in duality. Recently, in (Aubin-Frankowski,2021), we found new reproducing kernels in Linear-Quadratic optimal control by focusing on the Hilbert space of controlled trajectories, allowing for a convenient handling of state constraints and meeting points. We now extend this viewpoint to estimation problems where it is known that kernels are the covariances of stochastic processes. Here, the Markovian Gaussian processes stem from the linear stochastic differential equations describing the continuous-time dynamics and observations. Taking extensive care to require minimal invertibility requirements on the operators, we give novel explicit formulas for these covariances. We also determine their reproducing kernel Hilbert spaces, stressing the symmetries between a space of forward-time trajectories and a space of backward-time information vectors. The two spaces play an analogue role for filtering to Sobolev spaces in variational analysis, and allow to recover the Kalman estimate through a direct variational argument. For comparison, we then recover the Kalman filter and smoother formulas through more classical arguments based on the innovation process. Extension to discrete-time observations or infinite-dimensional state, tough technical, would be straightforward.

This paper tackles the problem of nonlinear systems, with sublinear growth but unbounded control, under perturbation of some time-varying state constraints. It is shown that, given a trajectory to be approximated, one can find a neighboring one that lies in the interior of the constraints, and which can be made arbitrarily close to the reference trajectory both in $L^\infty$-distance and $L^2$-control cost. This result is an important tool to prove the convergence of approximation schemes of state constraints based on interior solutions and is applicable to control-affine systems.

The Linear Quadratic Regulator (LQR), which is arguably the most classical problem in control theory, was recently related to kernel methods in (Aubin-Frankowski, SICON, 2021) for finite dimensional systems. We show that this result extends to infinite dimensional systems, i.e.\ control of linear partial differential equations. The quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories, for which we obtain a concise formula based on the solution of the differential Riccati equation. This paves the way to applying representer theorems from kernel methods to solve infinite dimensional optimal control problems.

Many problems in machine learning can be formulated as optimizing a convex functional over a vector space of measures. This paper studies the convergence of the mirror descent algorithm in this infinite-dimensional setting. Defining Bregman divergences through directional derivatives, we derive the convergence of the scheme for relatively smooth and convex pairs of functionals. Such assumptions allow to handle non-smooth functionals such as the Kullback--Leibler (KL) divergence. Applying our result to joint distributions and KL, we show that Sinkhorn's primal iterations for entropic optimal transport in the continuous setting correspond to a mirror descent, and we obtain a new proof of its (sub)linear convergence. We also show that Expectation Maximization (EM) can always formally be written as a mirror descent. When optimizing only on the latent distribution while fixing the mixtures parameters -- which corresponds to the Richardson--Lucy deconvolution scheme in signal processing -- we derive sublinear rates of convergence.

Hilbertian kernel methods and their positive semidefinite kernels have been extensively used in various fields of applied mathematics and machine learning, owing to their several equivalent characterizations. We here unveil an analogy with concepts from tropical geometry, proving that tropical positive semidefinite kernels are also endowed with equivalent viewpoints, stemming from Fenchel-Moreau conjugations. This tropical analogue of Aronszajn's theorem shows that these kernels correspond to a feature map, define monotonous operators, and generate max-plus function spaces endowed with a reproducing property. They furthermore include all the Hilbertian kernels classically studied as well as Monge arrays. However, two relevant notions of tropical reproducing kernels must be distinguished, based either on linear or sesquilinear interpretations. The sesquilinear interpretation is the most expressive one, since reproducing spaces then encompass classical max-plus spaces, such as those of (semi)convex functions. In contrast, in the linear interpretation, the reproducing kernels are characterized by a restrictive condition, von Neumann regularity. Finally, we provide a tropical analogue of the ``representer theorems'', showing that a class of infinite dimensional regression and interpolation problems admit solutions lying in finite dimensional spaces. We illustrate this theorem by an application to optimal control, in which tropical kernels allow one to represent the value function.

Among dissimilarities between probability distributions, the Kernel Stein Discrepancy (KSD) has received much interest recently. We investigate the properties of its Wasserstein gradient flow to approximate a target probability distribution $\pi$ on $\mathbb{R}^d$, known up to a normalization constant. This leads to a straightforwardly implementable, deterministic score-based method to sample from $\pi$, named KSD Descent, which uses a set of particles to approximate $\pi$. Remarkably, owing to a tractable loss function, KSD Descent can leverage robust parameter-free optimization schemes such as L-BFGS; this contrasts with other popular particle-based schemes such as the Stein Variational Gradient Descent algorithm. We study the convergence properties of KSD Descent and demonstrate its practical relevance. However, we also highlight failure cases by showing that the algorithm can get stuck in spurious local minima.

Shape constraints, such as non-negativity, monotonicity, convexity or supermodularity, play a key role in various applications of machine learning and statistics. However, incorporating this side information into predictive models in a hard way (for example at all points of an interval) for rich function classes is a notoriously challenging problem. We propose a unified and modular convex optimization framework, relying on second-order cone (SOC) tightening, to encode hard affine SDP constraints on function derivatives, for models belonging to vector-valued reproducing kernel Hilbert spaces (vRKHSs). The modular nature of the proposed approach allows to simultaneously handle multiple shape constraints, and to tighten an infinite number of constraints into finitely many. We prove the convergence of the proposed scheme and that of its adaptive variant, leveraging geometric properties of vRKHSs. Due to the covering-based construction of the tightening, the method is particularly well-suited to tasks with small to moderate input dimensions. The efficiency of the approach is illustrated in the context of shape optimization, safety-critical control, robotics and econometrics.

In this study, we provide an interpretation of the dual differential Riccati equation of Linear-Quadratic (LQ) optimal control problems. Adopting a novel viewpoint, we show that LQ optimal control can be seen as a regression problem over the space of controlled trajectories, and that the latter has a very natural structure as a reproducing kernel Hilbert space (RKHS). The dual Riccati equation then describes the evolution of the values of the LQ reproducing kernel when the initial time changes. This unveils new connections between control theory and kernel methods, a field widely used in machine learning.

The linear quadratic regulator problem is central in optimal control and was investigated since the very beginning of control theory. Nevertheless, when it includes affine state constraints, it remains very challenging from the classical ``maximum principle`` perspective. In this study we present how matrix-valued reproducing kernels allow for an alternative viewpoint. We show that the quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories. Drawing upon kernel formalism, we introduce a strengthened continuous-time convex optimization problem which can be tackled exactly with finite dimensional solvers, and which solution is interior to the constraints. When refining a time-discretization grid, this solution can be made arbitrarily close to the solution of the state-constrained Linear Quadratic Regulator. We illustrate the implementation of this method on a path-planning problem.

Shape constraints (such as non-negativity, monotonicity, convexity) play a central role in a large number of applications, as they usually improve performance for small sample size and help interpretability. However enforcing these shape requirements in a hard fashion is an extremely challenging problem. Classically, this task is tackled (i) in a soft way (without out-of-sample guarantees), (ii) by specialized transformation of the variables on a case-by-case basis, or (iii) by using highly restricted function classes, such as polynomials or polynomial splines. In this paper, we prove that hard affine shape constraints on function derivatives can be encoded in kernel machines which represent one of the most flexible and powerful tools in machine learning and statistics. Particularly, we present a tightened second-order cone constrained reformulation, that can be readily implemented in convex solvers. We prove performance guarantees on the solution, and demonstrate the efficiency of the approach in joint quantile regression with applications to economics and to the analysis of aircraft trajectories, among others., Comment: camera-ready paper

This study shows how Aronszajn's theory of reproducing kernels can be of use for the construction the Hilbert spaces of quantum theory. We show that the Feynman propagator is an example of a reproducing kernel under a boundedness condition. To every Lagrangian thus corresponds a Hilbert space that does not need to be postulated \emph{a priori}. For the free non-relativistic particle, we justify mathematically the concept of space-time granularity. Reproducing kernels allow for a functional, rather than distributional, description of the Hilbert spaces of quantum theory, including the Fock space., Comment: The proof of Lemma 2.2 contains an error