We give a simple approximation algorithm for a common generalization of many previously studied extensions of the stable matching problem with ties. These generalizations include the existence of critical vertices in the graph, amongst whom we must match as much as possible, free edges, that cannot be blocking edges and $\Delta$-stabilities, which mean that for an edge to block, the improvement should be large enough on one or both sides. We also introduce other notions to generalize these even further, which allows our framework to capture many existing and future applications. We show that our edge duplicating technique allows us to treat these different types of generalizations simultaneously, while also making the algorithm, the proofs and the analysis much simpler and shorter then in previous approaches. In particular, we answer an open question by Askalidis et al. (2013) about the existence of a $\frac{3}{2}$-approximation algorithm for the Max-SMTI problem with free edges. This demonstrates well that this technique can grasp the underlying essence of these problems quite well and have the potential to be able to solve countless future applications as well.
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This paper investigates the relationship between the universal approximation property of deep neural networks and topological characteristics of datasets. Our primary contribution is to introduce data topology-dependent upper bounds on the network width. Specifically, we first show that a three-layer neural network, applying a ReLU activation function and max pooling, can be designed to approximate an indicator function over a compact set, one that is encompassed by a tight convex polytope. This is then extended to a simplicial complex, deriving width upper bounds based on its topological structure. Further, we calculate upper bounds in relation to the Betti numbers of select topological spaces. Finally, we prove the universal approximation property of three-layer ReLU networks using our topological approach. We also verify that gradient descent converges to the network structure proposed in our study.
Most reinforcement learning algorithms treat the context under which they operate as a stationary, isolated and undisturbed environment. However, in the real world, the environment is constantly changing due to a variety of external influences. To address this problem, we study Markov Decision Processes (MDP) under the influence of an external temporal process. We formalize this notion and discuss conditions under which the problem becomes tractable with suitable solutions. We propose a policy iteration algorithm to solve this problem and theoretically analyze its performance.
We extend the powerful Pullback-Pushout (PBPO) approach for graph rewriting with strong matching. Our approach, called PBPO+, allows more control over the embedding of the pattern in the host graph, which is important for a large class of rewrite systems. We argue that PBPO+ can be considered a unifying theory in the general setting of quasitoposes, by demonstrating that PBPO+ can define a strict superset of the rewrite relations definable by PBPO, AGREE and DPO. Additionally, we show that PBPO+ is well suited for rewriting labeled graphs and some classes of attributed graphs, by introducing a lattice structure on the label set and requiring graph morphisms to be order-preserving.
DPPs were introduced by Macchi as a model in quantum optics the 1970s. Since then, they have been widely used as models and subsampling tools in statistics and computer science. Most applications require sampling from a DPP, and given their quantum origin, it is natural to wonder whether sampling a DPP on a quantum computer is easier than on a classical one. We focus here on DPPs over a finite state space, which are distributions over the subsets of $\{1,\dots,N\}$ parametrized by an $N\times N$ Hermitian kernel matrix. Vanilla sampling consists in two steps, of respective costs $\mathcal{O}(N^3)$ and $\mathcal{O}(Nr^2)$ operations on a classical computer, where $r$ is the rank of the kernel matrix. A large first part of the current paper consists in explaining why the state-of-the-art in quantum simulation of fermionic systems already yields quantum DPP sampling algorithms. We then modify existing quantum circuits, and discuss their insertion in a full DPP sampling pipeline that starts from practical kernel specifications. The bottom line is that, with $P$ (classical) parallel processors, we can divide the preprocessing cost by $P$ and build a quantum circuit with $\mathcal{O}(Nr)$ gates that sample a given DPP, with depth varying from $\mathcal{O}(N)$ to $\mathcal{O}(r\log N)$ depending on qubit-communication constraints on the target machine. We also connect existing work on the simulation of superconductors to Pfaffian point processes, which generalize DPPs and would be a natural addition to the machine learner's toolbox. Finally, the circuits are empirically validated on a classical simulator and on 5-qubit machines.
Actor-critic (AC) methods are widely used in reinforcement learning (RL) and benefit from the flexibility of using any policy gradient method as the actor and value-based method as the critic. The critic is usually trained by minimizing the TD error, an objective that is potentially decorrelated with the true goal of achieving a high reward with the actor. We address this mismatch by designing a joint objective for training the actor and critic in a decision-aware fashion. We use the proposed objective to design a generic, AC algorithm that can easily handle any function approximation. We explicitly characterize the conditions under which the resulting algorithm guarantees monotonic policy improvement, regardless of the choice of the policy and critic parameterization. Instantiating the generic algorithm results in an actor that involves maximizing a sequence of surrogate functions (similar to TRPO, PPO) and a critic that involves minimizing a closely connected objective. Using simple bandit examples, we provably establish the benefit of the proposed critic objective over the standard squared error. Finally, we empirically demonstrate the benefit of our decision-aware actor-critic framework on simple RL problems.
We review different (reduced) models for thin structures using bending as principal mechanism to undergo large deformations. Each model consists in the minimization of a fourth order energy, potentially subject to a nonconvex constraint. Equilibrium deformations are approximated using local discontinuous Galerkin (LDG) finite elements. The design of the discrete energies relies on a discrete Hessian operator defined on discontinuous functions with better approximation properties than the piecewise Hessian. Discrete gradient flows are put in place to drive the minimization process. They are chosen for their robustness and ability to preserve the nonconvex constraint. Several numerical experiments are presented to showcase the large variety of shapes that can be achieved with these models.
We consider the adversarial linear contextual bandit setting, which allows for the loss functions associated with each of $K$ arms to change over time without restriction. Assuming the $d$-dimensional contexts are drawn from a fixed known distribution, the worst-case expected regret over the course of $T$ rounds is known to scale as $\tilde O(\sqrt{Kd T})$. Under the additional assumption that the density of the contexts is log-concave, we obtain a second-order bound of order $\tilde O(K\sqrt{d V_T})$ in terms of the cumulative second moment of the learner's losses $V_T$, and a closely related first-order bound of order $\tilde O(K\sqrt{d L_T^*})$ in terms of the cumulative loss of the best policy $L_T^*$. Since $V_T$ or $L_T^*$ may be significantly smaller than $T$, these improve over the worst-case regret whenever the environment is relatively benign. Our results are obtained using a truncated version of the continuous exponential weights algorithm over the probability simplex, which we analyse by exploiting a novel connection to the linear bandit setting without contexts.
The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.
The Bayesian paradigm has the potential to solve core issues of deep neural networks such as poor calibration and data inefficiency. Alas, scaling Bayesian inference to large weight spaces often requires restrictive approximations. In this work, we show that it suffices to perform inference over a small subset of model weights in order to obtain accurate predictive posteriors. The other weights are kept as point estimates. This subnetwork inference framework enables us to use expressive, otherwise intractable, posterior approximations over such subsets. In particular, we implement subnetwork linearized Laplace: We first obtain a MAP estimate of all weights and then infer a full-covariance Gaussian posterior over a subnetwork. We propose a subnetwork selection strategy that aims to maximally preserve the model's predictive uncertainty. Empirically, our approach is effective compared to ensembles and less expressive posterior approximations over full networks.
Over the past few years, we have seen fundamental breakthroughs in core problems in machine learning, largely driven by advances in deep neural networks. At the same time, the amount of data collected in a wide array of scientific domains is dramatically increasing in both size and complexity. Taken together, this suggests many exciting opportunities for deep learning applications in scientific settings. But a significant challenge to this is simply knowing where to start. The sheer breadth and diversity of different deep learning techniques makes it difficult to determine what scientific problems might be most amenable to these methods, or which specific combination of methods might offer the most promising first approach. In this survey, we focus on addressing this central issue, providing an overview of many widely used deep learning models, spanning visual, sequential and graph structured data, associated tasks and different training methods, along with techniques to use deep learning with less data and better interpret these complex models --- two central considerations for many scientific use cases. We also include overviews of the full design process, implementation tips, and links to a plethora of tutorials, research summaries and open-sourced deep learning pipelines and pretrained models, developed by the community. We hope that this survey will help accelerate the use of deep learning across different scientific domains.
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