Kernels for sequentially ordered data

We present a novel framework for learning with sequential data of any kind, such as multivariate time series, strings, or sequences of graphs. The main result is a ”sequentialization” that transforms any kernel on a given domain into a kernel for sequences in that domain. This procedure preserves properties such as positive definiteness, the associated kernel feature map is an ordered variant of sample (cross-)moments, and this sequentialized kernel is consistent in the sense that it converges to a kernel for paths if sequences converge to paths (by discretization). Further, classical kernels for sequences arise as special cases of this method. We use dynamic programming and low-rank techniques for tensors to provide efficient algorithms to compute this sequentialized kernel

The Signature Kernel is the solution of a Goursat PDE

Recently, there has been an increased interest in the development of kernel methods for learning with sequential data. The signature kernel is a learning tool with potential to handle irregularly sampled, multivariate time series. In "Kernels for sequentially ordered data" the authors introduced a kernel trick for the truncated version of this kernel avoiding the exponential complexity that would have been involved in a direct computation. Here we show that for continuously differentiable paths, the signature kernel solves a hyperbolic PDE and recognize the connection with a well known class of differential equations known in the literature as Goursat problems. This Goursat PDE only depends on the increments of the input sequences, does not require the explicit computation of signatures and can be solved efficiently using state-of-the-arthyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from "Kernels for sequentially ordered data", but with the advantage that the PDE numerical scheme is well suited for GPU parallelization, which effectively reduces the complexity by a full order of magnitude in the length of the input sequences. In addition, we extend the previous analysis to the space of geometric rough paths and establish, using classical results from rough path theory, that the rough version of the signature kernel solves a rough integral equation analogous to the aforementioned Goursat PDE. Finally, we empirically demonstrate the effectiveness of our PDE kernel as a machine learning tool in various machine learning applications dealing with sequential data. We release the library sigkernel publicly available at https://github.com/crispitagorico/sigkernel

Run Time Approximation of Non-blocking Service Rates for Streaming Systems

Stream processing is a compute paradigm that promises safe and efficient parallelism. Modern big-data problems are often well suited for stream processing's throughput-oriented nature. Realization of efficient stream processing requires monitoring and optimization of multiple communications links. Most techniques to optimize these links use queueing network models or network flow models, which require some idea of the actual execution rate of each independent compute kernel within the system. What we want to know is how fast can each kernel process data independent of other communicating kernels. This is known as the "service rate" of the kernel within the queueing literature. Current approaches to divining service rates are static. Modern workloads, however, are often dynamic. Shared cloud systems also present applications with highly dynamic execution environments (multiple users, hardware migration, etc.). It is therefore desirable to continuously re-tune an application during run time (online) in response to changing conditions. Our approach enables online service rate monitoring under most conditions, obviating the need for reliance on steady state predictions for what are probably non-steady state phenomena. First, some of the difficulties associated with online service rate determination are examined. Second, the algorithm to approximate the online non-blocking service rate is described. Lastly, the algorithm is implemented within the open source RaftLib framework for validation using a simple microbenchmark as well as two full streaming applications.Comment: technical repor

Learning to Race through Coordinate Descent Bayesian Optimisation

In the automation of many kinds of processes, the observable outcome can often be described as the combined effect of an entire sequence of actions, or controls, applied throughout its execution. In these cases, strategies to optimise control policies for individual stages of the process might not be applicable, and instead the whole policy might have to be optimised at once. On the other hand, the cost to evaluate the policy's performance might also be high, being desirable that a solution can be found with as few interactions as possible with the real system. We consider the problem of optimising control policies to allow a robot to complete a given race track within a minimum amount of time. We assume that the robot has no prior information about the track or its own dynamical model, just an initial valid driving example. Localisation is only applied to monitor the robot and to provide an indication of its position along the track's centre axis. We propose a method for finding a policy that minimises the time per lap while keeping the vehicle on the track using a Bayesian optimisation (BO) approach over a reproducing kernel Hilbert space. We apply an algorithm to search more efficiently over high-dimensional policy-parameter spaces with BO, by iterating over each dimension individually, in a sequential coordinate descent-like scheme. Experiments demonstrate the performance of the algorithm against other methods in a simulated car racing environment.Comment: Accepted as conference paper for the 2018 IEEE International Conference on Robotics and Automation (ICRA