In supervised learning, we seek to learn to predict a target variable from data. The way this happens is by having access to a number of data vectors with their associated targets (i.e. labels or physical measurements), defining some functional form that maps data to targets, and a merit criterion which is small when estimates are close to targets. One salient question for this domain is how to choose the functional form such that the resulting optimization problem is solvable, but sufficiently general so as to be able to capture important relationships between data and targets. In this thread of research, we investigate how to manage this tradeoff for the case that estimators belong to a Reproducing Kernel Hilbert Space (RKHS), which yields a very rich class of estimators. However, owing to the Representer Theorem, this choice also makes the complexity of the optimization problem comparable to the training sample size, which is untenable for streaming applications or large-scale problems. However, by carefully constructing subspaces onto which we project RKHS-valued stochastic gradient algorithms, we are able to find optimally compressed kernelized regression functions. This is in contrast to a long history of literature trying to ameliorate the complexity growth of kernel methods with the training sample size that typically violate the optimality properties of the iterative sequences to which they’re applied.

We call this method Parsimonious Online Learning with Kernels (POLK), and illustrate a couple examples of the decision surface defined by POLK when used with a Gaussian kernel for two-dimensional multi-class classification. On the left we visualize the surface defined by kernel logistic regression; on the right a kernel support vector machine classifier is visualized. [Video]

Currently I am investigating extensions of these ideas that (1) overcome the bias-variance tradeoff; and (2) how to scale up this behavior to high parameter spaces such as vision or acoustics that have hierarchical structure through use of convolutional kernels.

In reinforcement learning, an autonomous agent observes a sequence of rewards as it traverses its environment based upon the actions it takes. Depending on which actions it takes, it may end up in more or less rewarding states. The question is how to choose the policy, i.e., the map from states to actions, so as to ensure the long-run (discounted) accumulation of rewards is maximized. This long-run accumulation is called the value. Classical analysis on this topic has shown that it’s possible to reformulate the problem of both policy evaluation and policy learning as different function-valued fixed point problems called Bellman equations if the state and action spaces are continuous. Thus, many classical techniques for RL use stochastic fixed point iterations, e.g., Temporal Difference Learning and Q Learning. These methods break down when combined with function approximation, however, which is intrinsically required for continuous Markov Decision Problems. My approach to mitigating these issues is to reformulate the fixed point problems defined by Bellman’s evaluation into compositional optimization problems, for which it is possible to develop iterative algorithms that can be shown to have globally stable Lyapunov functions. This is done by allowing the reproducing Kernel Hilbert space to define the value function parameterization. See an example of how this works on the Mountain Car domain in the below plot, where we should that the Bellman error converges and we obtain a memory-efficient value function that is transparently interpretable in the contour plot. We handle the growth of the kernel parameterization through sparse subspace projections in a manner very similar to POLK.

This approach may be adapted to solve the Bellman optimality equation for action-value functions. In the below left contour plot we show how this may be used to solve the Continuous Mountain car and obtain interpretable results. On the left below, the value function associated with the learned policy that is obtained by maximizing the action-value function. On the right we plot the resulting policy.

Currently I building up tools to address shortcomings in our existing compositional optimization approach to solving Bellman’s optimality equation to address the fact that it requires smooth function approximation of a (1) non-differentiable function and (2) attempting to circumvent the need for nested expectations by better understanding how stochastic fixed point methods can be combined with function approximation.

Many machine learning and signal processing problems may be formulated as convex optimization problems with an objective that may be written as the sum of predictive risks evaluated at each sample point of a training data set. This is variously referred to as empirical risk minimization or stochastic average approximation. When the number of sample points is static and not too large, deterministic iterative algorithms such as gradient descent or Newton's method provide effective tools for solviong this class of problems. When the number of sample points is huge-scale, possible infinite, stochastic approximation becomes necessary. Stochastic gradient methods operate on stochastic gradients in lieu of true gradients, which allow for processing data points one at a time, and under suitable conditions converge to the minimum of an expected risk over all data realizations. This methodology has been around since a landmark paper of Robbins and Monroe in 1951, but is increasingly popular in the "Big Data" era. Within this problem class, I have worked on how to handle learning problems where the number of predictive parameters, or features, is comparable to the (huge-scale) number of sample points. I have developed a parallelized doubly stochastic approximation algorithms, which allows for operating on random subsets of both sample points and features. It does so by combing the merits of stochastic gradient method with random coordinate descent.

Multi-agent systems refer to settings in which distinct computing nodes that may communicate with one another according to some network structure aim to solve a task which is global to the network. Sensor networks, computers networks, and robotic teams may all be described with this perspective. Classically, this problem class has led to the development of decentralized optimization algorithms, where distinct nodes operate on local information only. Through appropriate  information mixing strategies, agents may learn a decision variable which is optimal in terms of observations aggregated globally over the network. 

My contributions to this problem domain are in developing new information mixing strategies. Prior approaches are based upon schemes where agents combine a weighted average of neighbors decision variables with a local gradient step ("the consensus protocol"), whereas the method I propose for this setting, called the saddle point algorithm, expoits duality in convex optimization theory to have agents approximately enforce local agreement constraints while learning based upon local information. This saddle point method can solve more general classes of decentralized optimization problems than consensus methods, i.e. problems in which each agent's cost function is parameterized by a random variable with a possibly distinct distribution function, or more general types of network constraints. Moreover, these primal-dual algorithms are well-suited to online or dynamic settings where training observations become sequentially available, and agents must repeatedly adapt to new information. [Video]

networkonline learning in networks

Robotics is an exciting field for signal processing researchers because there are so many frontiers to which we may explore and contribute. My particular focus has been on learning-based control in single-robot and multi-robot systems. In particular, I study information processing strategies for streaming sensory data, such that useful intelligence may be extracted from the robot's operating domain for adapting its control strategies.This problem breaks down into three components (1) how to train a statistical model online relating sensory data to some label or output variable space, (2) how to map the predictions of this statistical model into the control space, and (3) tailoring the robotic control strategy to the statistics of its operating environment.


For problem (1), I have developed extensions of task-driven dictionary learning to decentralized settings. Task-driven dictionary learning is a method for learning a signal representation and statistical model relating a feature space and predictive variable jointly in a manner that is tuned for minimizing the expected prediction risk. By creating a framework for this task in multi-agent systems, I've provided the capability for robotic networks to handle streaming data and collaboratively exchange information to maximize the informativeness of the paths they've traversed for a particular prediction task.

The task of how to map statistical predictions into the control space of the autonomous system (2) classically has been treated with robust control techniques, in which one adds chance constraints or stochastic variables to the objective of an optimal control formulation. While doing yields robust performance for linear systems, most robotic kinematics are highly nonlinear, in which case tried and true methods for linear systems are not appropriate. To circumvent this issue, I am considering developing new formulations of receding horizon control of nonlinear systems where the cost functional automatically switches between optimal and robust control based on the level of predicted uncertainty in the environment. By approaching task (3) in this manner, I am developing a method such that an autonomous system may learn online to compensate for the simplifications made in kinematic models of robot platforms that are typically used to generate control signals, and thus have a robot adjust in real-time to unexpected changes in the environment in which it is operating. By learning the uncertainty in a control action, we may use this estimate to correct for deviations between ground truth and some kinematic model. In the below figure, the red path is the one that uses uncertainty estimated by simple averaging; the green path is the ground truth path taken by the robot; and the blue path is the result of our uncertainty estimate. On the left we picture these paths before learning whereas on the right we picture them after learning. After learning, they match very closely.

Prior to coming to UPenn, the unifying theme of my research projects was Monte Carlo and stochastic diffusion approximation algorithms. My undergraduate research project in the Math Dept. at Washington University in St. Louis was to study probabilistic models of predator-prey interactions in migrational settings. To do so, I made use of tools in stochastic simulation and diffusion approximations to discern under which cases population stability, explosion, and extinction occurred across different sites in a nonlinear networked dynamical system.

I spent 2011-2012  at the U.S. Army Research Laboratory ALC as part of the Science Outreach for Army Research (SOAR) program investigating different frameworks for modeling robotic networks in complex domains. The first project I worked on was to evaluate a "bio--inspired" control strategy on an asynchronous parallel computing architecture. Numerically I was able to conclude that if the amount of information processed passed a certain threshold, the system was controllable. The other project I worked on was generalizing a leader-follower framework for a robotic network to maintain communications connectivity while exploring the environment. I implemented a simulated annealing strategy to overcome the tendency of follower-robots becoming trapped at obstacles in the environment, which is a consequence of non-convexity of the feasible space.

I also worked as a biostatistical research intern at Washington University School of Medicine's Summer Institute for Training in Biostatistics (SIBS), where I performed statistical analyses and explored latent variable patterns in clinical trials via independent component analysis, and  presented my findings to a team of physicians.