We present a novel methodology to jointly perform multi-task learning and infer intrinsic relationship among tasks by an interpretable and sparse graph. Unlike existing multi-task learning methodologies, the graph structure is not assumed to be known a priori or estimated separately in a preprocessing step. Instead, our graph is learned simultaneously with model parameters of each task, thus it reflects the critical relationship among tasks in the specific prediction problem. We characterize graph structure with its weighted adjacency matrix and show that the overall objective can be optimized alternatively until convergence. We also show that our methodology can be simply extended to a nonlinear form by being embedded into a multi-head radial basis function network (RBFN). Extensive experiments, against six state-of-the-art methodologies, on both synthetic data and real-world applications suggest that our methodology is able to reduce generalization error, and, at the same time, reveal a sparse graph over tasks that is much easier to interpret.

We present an approach to construct appropriate and efficient emulators for Hamiltonian flow maps. Intended future applications are long-term tracing of fast charged particles in accelerators and magnetic plasma confinement configurations. The method is based on multi-output Gaussian process regression on scattered training data. To obtain long-term stability the symplectic property is enforced via the choice of the matrix-valued covariance function. Based on earlier work on spline interpolation we observe derivatives of the generating function of a canonical transformation. A product kernel produces an accurate implicit method, whereas a sum kernel results in a fast explicit method from this approach. Both correspond to a symplectic Euler method in terms of numerical integration. These methods are applied to the pendulum and the H\'enon-Heiles system and results compared to an symmetric regression with orthogonal polynomials. In the limit of small mapping times, the Hamiltonian function can be identified with a part of the generating function and thereby learned from observed time-series data of the system's evolution. Besides comparable performance of implicit kernel and spectral regression for symplectic maps, we demonstrate a substantial increase in performance for learning the Hamiltonian function compared to existing approaches.

How can we update data for a machine learning model after it has already trained on that data? In this paper, we introduce DART, a variant of random forests that supports adding and removing training data with minimal retraining. Data updates in DART are exact, meaning that adding or removing examples from a DART model yields exactly the same model as retraining from scratch on updated data. DART uses two techniques to make updates efficient. The first is to cache data statistics at each node and training data at each leaf, so that only the necessary subtrees are retrained. The second is to choose the split variable randomly at the upper levels of each tree, so that the choice is completely independent of the data and never needs to change. At the lower levels, split variables are chosen to greedily maximize a split criterion such as Gini index or mutual information. By adjusting the number of random-split levels, DART can trade off between more accurate predictions and more efficient updates. In experiments on ten real-world datasets and one synthetic dataset, we find that DART is orders of magnitude faster than retraining from scratch while sacrificing very little in terms of predictive performance.

When machine learning supports decision-making in safety-critical systems, it is important to verify and understand the reasons why a particular output is produced. Although feature importance calculation approaches assist in interpretation, there is a lack of consensus regarding how features' importance is quantified, which makes the explanations offered for the outcomes mostly unreliable. A possible solution to address the lack of agreement is to combine the results from multiple feature importance quantifiers to reduce the variance of estimates. Our hypothesis is that this will lead to more robust and trustworthy interpretations of the contribution of each feature to machine learning predictions. To assist test this hypothesis, we propose an extensible Framework divided in four main parts: (i) traditional data pre-processing and preparation for predictive machine learning models; (ii) predictive machine learning; (iii) feature importance quantification and (iv) feature importance decision fusion using an ensemble strategy. We also introduce a novel fusion metric and compare it to the state-of-the-art. Our approach is tested on synthetic data, where the ground truth is known. We compare different fusion approaches and their results for both training and test sets. We also investigate how different characteristics within the datasets affect the feature importance ensembles studied. Results show that our feature importance ensemble Framework overall produces 15% less feature importance error compared to existing methods. Additionally, results reveal that different levels of noise in the datasets do not affect the feature importance ensembles' ability to accurately quantify feature importance, whereas the feature importance quantification error increases with the number of features and number of orthogonal informative features.

Signal Temporal Logic (STL) [14] is gaining momentum as a requirement specification language for complex systems and, in particular, Cyber-Physical Systems [4]. STL has been applied in several flavours, from Runtime-monitoring [4] to control synthesis [10] and falsification problems [9], and recently also within learning algorithms, trying to find a maximally discriminating formula between sets of trajectories [5, 3, 16]. In these applications, a central role is played by the real-valued quantitative semantics [8], measuring robustness of satisfaction. Most of the applications of STL have been applied to deterministic (hybrid) systems, with less emphasis on non-deterministic or stochastic ones [2]. Another area in which formal methods are providing interesting tools is in logic-based distances between models, like bisimulation metrics for Markov models [1], which are typically based on a branching logic. In fact, extending these ideas to linear time logic is hard [7], and typically requires statistical approximations.

Interpretable Multi-Task Learning can be expressed as learning a sparse graph of the task relationship based on the prediction performance of the learned models. Since many natural phenomenon exhibit sparse structures, enforcing sparsity on learned models reveals the underlying task relationship. Moreover, different sparsification degrees from a fully connected graph uncover various types of structures, like cliques, trees, lines, clusters or fully disconnected graphs. In this paper, we propose a bilevel formulation of multi-task learning that induces sparse graphs, thus, revealing the underlying task relationships, and an efficient method for its computation. We show empirically how the induced sparse graph improves the interpretability of the learned models and their relationship on synthetic and real data, without sacrificing generalization performance.

The history of deep learning has shown that human-designed problem-specific networks can greatly improve the classification performance of general neural models. In most practical cases, however, choosing the optimal architecture for a given task remains a challenging problem. Recent architecture-search methods are able to automatically build neural models with strong performance but fail to fully appreciate the interaction between neural architecture and weights. This work investigates the problem of disentangling the role of the neural structure and its edge weights, by showing that well-trained architectures may not need any link-specific fine-tuning of the weights. We compare the performance of such weight-free networks (in our case these are binary networks with {0, 1}-valued weights) with random, weight-agnostic, pruned and standard fully connected networks. To find the optimal weight-agnostic network, we use a novel and computationally efficient method that translates the hard architecture-search problem into a feasible optimization problem.More specifically, we look at the optimal task-specific architectures as the optimal configuration of binary networks with {0, 1}-valued weights, which can be found through an approximate gradient descent strategy. Theoretical convergence guarantees of the proposed algorithm are obtained by bounding the error in the gradient approximation and its practical performance is evaluated on two real-world data sets. For measuring the structural similarities between different architectures, we use a novel spectral approach that allows us to underline the intrinsic differences between real-valued networks and weight-free architectures.

The notion of generalization in classical Statistical Learning is often attached to the postulate that data points are independent and identically distributed (IID) random variables. While relevant in many applications, this postulate may not hold in general, encouraging the development of learning frameworks that are robust to non-IID data. In this work, we consider the regression problem from an Optimal Recovery perspective. Relying on a model assumption comparable to choosing a hypothesis class, a learner aims at minimizing the worst-case error, without recourse to any probabilistic assumption on the data. We first develop a semidefinite program for calculating the worst-case error of any recovery map in finite-dimensional Hilbert spaces. Then, for any Hilbert space, we show that Optimal Recovery provides a formula which is user-friendly from an algorithmic point-of-view, as long as the hypothesis class is linear. Interestingly, this formula coincides with kernel ridgeless regression in some cases, proving that minimizing the average error and worst-case error can yield the same solution. We provide numerical experiments in support of our theoretical findings.

Decision forests are popular tools for classification and regression. These forests naturally produce proximity matrices measuring how often each pair of observations lies in the same leaf node. It has been demonstrated that these proximity matrices can be thought of as kernels, connecting the decision forest literature to the extensive kernel machine literature. While other kernels are known to have strong theoretical properties such as being characteristic, no similar result is available for any decision forest based kernel. In this manuscript, we prove that the decision forest induced proximity can be made characteristic, which can be used to yield a universally consistent statistic for testing independence. We demonstrate the performance of the induced kernel on a suite of 20 high-dimensional independence test settings. We also show how this learning kernel offers insights into relative feature importance. The decision forest induced kernel typically achieves substantially higher testing power than existing popular methods in statistical tests.

The Intel Distribution for Python (IDP), part of the Intel AI Analytics Toolkit, includes an optimized scikit-learn that accelerates a selection of common estimators (e.g., logistic regression, singular value decomposition, principal component analysis). These functions are built on top of the Intel Data Analytics Acceleration Library (DAAL) so they achieve performance close to that equivalent C programs. The DAAL-powered estimators are implemented in the daal4py package. DAAL's performance comes from efficient use of multiple CPU cores, cache-friendly blocking, and effective use of processor instruction sets. It is tuned to run best on Intel processors.