Global optimization of decision trees has shown to be promising in terms of accuracy, size, and consequently human comprehensibility. However, many of the methods used rely on general-purpose solvers for which scalability remains an issue. Dynamic programming methods have been shown to scale much better because they exploit the tree structure by solving subtrees as independent subproblems. However, this only works when an objective can be optimized separately for subtrees. We explore this relationship in detail and show necessary and sufficient conditions for such separability and generalize previous dynamic programming approaches into a framework that can optimize any combination of separable objectives and constraints. Experiments on four application domains show the general applicability of this framework, while outperforming the scalability of general-purpose solvers by a large margin.
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Having efficient testing strategies is a core challenge that needs to be overcome for the release of automated driving. This necessitates clear requirements as well as suitable methods for testing. In this work, the requirements for perception modules are considered with respect to relevance. The concept of relevance currently remains insufficiently defined and specified. In this paper, we propose a novel methodology to overcome this challenge by exemplary application to collision safety in the highway domain. Using this general system and use case specification, a corresponding concept for relevance is derived. Irrelevant objects are thus defined as objects which do not limit the set of safe actions available to the ego vehicle under consideration of all uncertainties. As an initial step, the use case is decomposed into functional scenarios with respect to collision relevance. For each functional scenario, possible actions of both the ego vehicle and any other dynamic object are formalized as equations. This set of possible actions is constrained by traffic rules, yielding relevance criteria. As a result, we present a conservative estimation which dynamic objects are relevant for perception and need to be considered for a complete evaluation. The estimation provides requirements which are applicable for offline testing and validation of perception components. A visualization is presented for examples from the highD dataset, showing the plausibility of the results. Finally, a possibility for a future validation of the presented relevance concept is outlined.
A key challenge in Bayesian decentralized data fusion is the `rumor propagation' or `double counting' phenomenon, where previously sent data circulates back to its sender. It is often addressed by approximate methods like covariance intersection (CI) which takes a weighted average of the estimates to compute the bound. The problem is that this bound is not tight, i.e. the estimate is often over-conservative. In this paper, we show that by exploiting the probabilistic independence structure in multi-agent decentralized fusion problems a tighter bound can be found using (i) an expansion to the CI algorithm that uses multiple (non-monolithic) weighting factors instead of one (monolithic) factor in the original CI and (ii) a general optimization scheme that is able to compute optimal bounds and fully exploit an arbitrary dependency structure. We compare our methods and show that on a simple problem, they converge to the same solution. We then test our new non-monolithic CI algorithm on a large-scale target tracking simulation and show that it achieves a tighter bound and a more accurate estimate compared to the original monolithic CI.
Rule-based models, such as decision trees, appeal to practitioners due to their interpretable nature. However, the learning algorithms that produce such models are often vulnerable to spurious associations and thus, they are not guaranteed to extract causally-relevant insights. In this work, we build on ideas from the invariant causal prediction literature to propose Invariant Causal Set Covering Machines, an extension of the classical Set Covering Machine algorithm for conjunctions/disjunctions of binary-valued rules that provably avoids spurious associations. We demonstrate both theoretically and empirically that our method can identify the causal parents of a variable of interest in polynomial time.
Information about action costs is critical for real-world AI planning applications. Rather than rely solely on declarative action models, recent approaches also use black-box external action cost estimators, often learned from data, that are applied during the planning phase. These, however, can be computationally expensive, and produce uncertain values. In this paper we suggest a generalization of deterministic planning with action costs that allows selecting between multiple estimators for action cost, to balance computation time against bounded estimation uncertainty. This enables a much richer -- and correspondingly more realistic -- problem representation. Importantly, it allows planners to bound plan accuracy, thereby increasing reliability, while reducing unnecessary computational burden, which is critical for scaling to large problems. We introduce a search algorithm, generalizing $A^*$, that solves such planning problems, and additional algorithmic extensions. In addition to theoretical guarantees, extensive experiments show considerable savings in runtime compared to alternatives.
Reinforcement learning (RL) algorithms have proven transformative in a range of domains. To tackle real-world domains, these systems often use neural networks to learn policies directly from pixels or other high-dimensional sensory input. By contrast, much theory of RL has focused on discrete state spaces or worst-case analysis, and fundamental questions remain about the dynamics of policy learning in high-dimensional settings. Here, we propose a solvable high-dimensional model of RL that can capture a variety of learning protocols, and derive its typical dynamics as a set of closed-form ordinary differential equations (ODEs). We derive optimal schedules for the learning rates and task difficulty - analogous to annealing schemes and curricula during training in RL - and show that the model exhibits rich behaviour, including delayed learning under sparse rewards; a variety of learning regimes depending on reward baselines; and a speed-accuracy trade-off driven by reward stringency. Experiments on variants of the Procgen game "Bossfight" and Arcade Learning Environment game "Pong" also show such a speed-accuracy trade-off in practice. Together, these results take a step towards closing the gap between theory and practice in high-dimensional RL.
Plug-and-play (PnP) denoising is a popular iterative framework for solving imaging inverse problems using off-the-shelf image denoisers. Their empirical success has motivated a line of research that seeks to understand the convergence of PnP iterates under various assumptions on the denoiser. While a significant amount of research has gone into establishing the convergence of the PnP iteration for different regularity conditions on the denoisers, not much is known about the asymptotic properties of the converged solution as the noise level in the measurement tends to zero, i.e., whether PnP methods are provably convergent regularization schemes under reasonable assumptions on the denoiser. This paper serves two purposes: first, we provide an overview of the classical regularization theory in inverse problems and survey a few notable recent data-driven methods that are provably convergent regularization schemes. We then continue to discuss PnP algorithms and their established convergence guarantees. Subsequently, we consider PnP algorithms with linear denoisers and propose a novel spectral filtering technique to control the strength of regularization arising from the denoiser. Further, by relating the implicit regularization of the denoiser to an explicit regularization functional, we rigorously show that PnP with linear denoisers leads to a convergent regularization scheme. More specifically, we prove that in the limit as the noise vanishes, the PnP reconstruction converges to the minimizer of a regularization potential subject to the solution satisfying the noiseless operator equation. The theoretical analysis is corroborated by numerical experiments for the classical inverse problem of tomographic image reconstruction.
Class imbalance poses new challenges when it comes to classifying data streams. Many algorithms recently proposed in the literature tackle this problem using a variety of data-level, algorithm-level, and ensemble approaches. However, there is a lack of standardized and agreed-upon procedures and benchmarks on how to evaluate these algorithms. This work proposes a standardized, exhaustive, and comprehensive experimental framework to evaluate algorithms in a collection of diverse and challenging imbalanced data stream scenarios. The experimental study evaluates 24 state-of-the-art data streams algorithms on 515 imbalanced data streams that combine static and dynamic class imbalance ratios, instance-level difficulties, concept drift, real-world and semi-synthetic datasets in binary and multi-class scenarios. This leads to a large-scale experimental study comparing state-of-the-art classifiers in the data stream mining domain. We discuss the advantages and disadvantages of state-of-the-art classifiers in each of these scenarios and we provide general recommendations to end-users for selecting the best algorithms for imbalanced data streams. Additionally, we formulate open challenges and future directions for this domain. Our experimental framework is fully reproducible and easy to extend with new methods. This way, we propose a standardized approach to conducting experiments in imbalanced data streams that can be used by other researchers to create complete, trustworthy, and fair evaluation of newly proposed methods. Our experimental framework can be downloaded from //github.com/canoalberto/imbalanced-streams.
We study the size of a neural network needed to approximate the maximum function over $d$ inputs, in the most basic setting of approximating with respect to the $L_2$ norm, for continuous distributions, for a network that uses ReLU activations. We provide new lower and upper bounds on the width required for approximation across various depths. Our results establish new depth separations between depth 2 and 3, and depth 3 and 5 networks, as well as providing a depth $\mathcal{O}(\log(\log(d)))$ and width $\mathcal{O}(d)$ construction which approximates the maximum function, significantly improving upon the depth requirements of the best previously known bounds for networks with linearly-bounded width. Our depth separation results are facilitated by a new lower bound for depth 2 networks approximating the maximum function over the uniform distribution, assuming an exponential upper bound on the size of the weights. Furthermore, we are able to use this depth 2 lower bound to provide tight bounds on the number of neurons needed to approximate the maximum by a depth 3 network. Our lower bounds are of potentially broad interest as they apply to the widely studied and used \emph{max} function, in contrast to many previous results that base their bounds on specially constructed or pathological functions and distributions.
Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.
Neural networks have shown tremendous growth in recent years to solve numerous problems. Various types of neural networks have been introduced to deal with different types of problems. However, the main goal of any neural network is to transform the non-linearly separable input data into more linearly separable abstract features using a hierarchy of layers. These layers are combinations of linear and nonlinear functions. The most popular and common non-linearity layers are activation functions (AFs), such as Logistic Sigmoid, Tanh, ReLU, ELU, Swish and Mish. In this paper, a comprehensive overview and survey is presented for AFs in neural networks for deep learning. Different classes of AFs such as Logistic Sigmoid and Tanh based, ReLU based, ELU based, and Learning based are covered. Several characteristics of AFs such as output range, monotonicity, and smoothness are also pointed out. A performance comparison is also performed among 18 state-of-the-art AFs with different networks on different types of data. The insights of AFs are presented to benefit the researchers for doing further research and practitioners to select among different choices. The code used for experimental comparison is released at: \url{//github.com/shivram1987/ActivationFunctions}.
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