Sequential modelling with self-attention has achieved cutting edge performances in natural language processing. With advantages in model flexibility, computation complexity and interpretability, self-attention is gradually becoming a key component in event sequence models. However, like most other sequence models, self-attention does not account for the time span between events and thus captures sequential signals rather than temporal patterns. Without relying on recurrent network structures, self-attention recognizes event orderings via positional encoding. To bridge the gap between modelling time-independent and time-dependent event sequence, we introduce a functional feature map that embeds time span into high-dimensional spaces. By constructing the associated translation-invariant time kernel function, we reveal the functional forms of the feature map under classic functional function analysis results, namely Bochner's Theorem and Mercer's Theorem. We propose several models to learn the functional time representation and the interactions with event representation. These methods are evaluated on real-world datasets under various continuous-time event sequence prediction tasks. The experiments reveal that the proposed methods compare favorably to baseline models while also capturing useful time-event interactions.

We study the robustness verification problem for tree based models, including decision trees, random forests (RFs) and gradient boosted decision trees (GBDTs). Formal robustness verification of decision tree ensembles involves finding the exact minimal adversarial perturbation or a guaranteed lower bound of it. Existing approaches find the minimal adversarial perturbation by a mixed integer linear programming (MILP) problem, which takes exponential time so is impractical for large ensembles. Although this verification problem is NP-complete in general, we give a more precise complexity characterization. We show that there is a simple linear time algorithm for verifying a single tree, and for tree ensembles the verification problem can be cast as a max-clique problem on a multi-partite graph with bounded boxicity. For low dimensional problems when boxicity can be viewed as constant, this reformulation leads to a polynomial time algorithm. For general problems, by exploiting the boxicity of the graph, we develop an efficient multi-level verification algorithm that can give tight lower bounds on robustness of decision tree ensembles, while allowing iterative improvement and any-time termination. On RF/GBDT models trained on 10 datasets, our algorithm is hundreds of times faster than a previous approach that requires solving MILPs, and is able to give tight robustness verification bounds on large GBDTs with hundreds of deep trees.

Graph matching plays a central role in such fields as computer vision, pattern recognition, and bioinformatics. Graph matching problems can be cast as two types of quadratic assignment problems (QAPs): Koopmans-Beckmann's QAP or Lawler's QAP. In our paper, we provide a unifying view for these two problems by introducing new rules for array operations in Hilbert spaces. Consequently, Lawler's QAP can be considered as the Koopmans-Beckmann's alignment between two arrays in reproducing kernel Hilbert spaces (RKHS), making it possible to efficiently solve the problem without computing a huge affinity matrix. Furthermore, we develop the entropy-regularized Frank-Wolfe (EnFW) algorithm for optimizing QAPs, which has the same convergence rate as the original FW algorithm while dramatically reducing the computational burden for each outer iteration. We conduct extensive experiments to evaluate our approach, and show that our algorithm significantly outperforms the state-of-the-art in both matching accuracy and scalability.

Modern machine learning systems are susceptible to adversarial examples; inputs which clearly preserve the characteristic semantics of a given class, but whose classification is (usually confidently) incorrect. Existing approaches to adversarial defense generally rely on modifying the input, e.g.

How might one "reduce" a graph? That is, generate a smaller graph that preserves the global structure at the expense of discarding local details? There has been extensive work on both graph sparsification (removing edges) and graph coarsening (merging nodes, often by edge contraction); however, these operations are currently treated separately. Interestingly, for a planar graph, edge deletion corresponds to edge contraction in its planar dual (and more generally, for a graphical matroid and its dual). Moreover, with respect to the dynamics induced by the graph Laplacian (e.g., diffusion), deletion and contraction are physical manifestations of two reciprocal limits: edge weights of 0 and 1, respectively. In this work, we provide a unifying framework that captures both of these operations, allowing one to simultaneously sparsify and coarsen a graph while preserving its large-scale structure. The limit of infinite edge weight is rarely considered, as many classical notions of graph similarity diverge.

In optimization the duality gap between the primal and the dual problems is a measure of the suboptimality of any primal-dual point. In classical mechanics the equations of motion of a system can be derived from the Hamiltonian function, which is a quantity that describes the total energy of the system. In this paper we consider a convex optimization problem consisting of the sum of two convex functions, sometimes referred to as a composite objective, and we identify the duality gap to be the'energy' of the system. In the Hamiltonian formalism the energy is conserved, so we add a contractive term to the standard equations of motion so that this energy decreases linearly (i.e., geometrically) with time. This yields a continuous-time ordinary differential equation (ODE) in the primal and dual variables which converges to zero duality gap, i.e., optimality.

Attention is all you need.

A computational problem in biological reward-based learning is how credit assignment is performed in the nucleus accumbens (NAc) to update synaptic weights. Much research suggests that NAc dopamine encodes temporal-difference (TD) errors for learning value predictions. However, dopamine is synchronously distributed in regionally homogeneous concentrations, which does not support explicit credit assignment (like used by backpropagation). It is unclear whether distributed errors alone are sufficient for synapses to make coordinated updates to learn complex, nonlinear reward-based learning tasks.

Designing rewards for Reinforcement Learning (RL) is challenging because it needs to convey the desired task, be efficient to optimize, and be easy to compute. The latter is particularly problematic when applying RL to robotics, where detecting whether the desired configuration is reached might require considerable supervision and instrumentation. Furthermore, we are often interested in being able to reach a wide range of configurations, hence setting up a different reward every time might be unpractical. Methods like Hindsight Experience Replay (HER) have recently shown promise to learn policies able to reach many goals, without the need of a reward. Unfortunately, without tricks like resetting to points along the trajectory, HER might require many samples to discover how to reach certain areas of the state-space. In this work we propose a novel algorithm goalGAIL, which incorporates demonstrations to drastically speed up the convergence to a policy able to reach any goal, surpassing the performance of an agent trained with other Imitation Learning algorithms. Furthermore, we show our method can also be used when the available expert trajectories do not contain the actions or when the expert is suboptimal, which makes it applicable when only kinesthetic, third-person or noisy demonstrations are available.

We study learning properties of accelerated gradient descent methods for linear least-squares in Hilbert spaces. We analyze the implicit regularization properties of Nesterov acceleration and a variant of heavy-ball in terms of corresponding learning error bounds. Our results show that acceleration can provides faster bias decay than gradient descent, but also suffers of a more unstable behavior. As a result acceleration cannot be in general expected to improve learning accuracy with respect to gradient descent, but rather to achieve the same accuracy with reduced computations. Our theoretical results are validated by numerical simulations. Our analysis is based on studying suitable polynomials induced by the accelerated dynamics and combining spectral techniques with concentration inequalities.