We consider distributed optimization problems where forming the Hessian is computationally challenging and communication is a significant bottleneck. We develop unbiased parameter averaging methods for randomized second order optimization that employ sampling and sketching of the Hessian. Existing works do not take the bias of the estimators into consideration, which limits their application to massively parallel computation. We provide closed-form formulas for regularization parameters and step sizes that provably minimize the bias for sketched Newton directions. We also extend the framework of second order averaging methods to introduce an unbiased distributed optimization framework for heterogeneous computing systems with varying worker resources. Additionally, we demonstrate the implications of our theoretical findings via large scale experiments performed on a serverless computing platform.

Machine learning is increasingly being used in various applications that make decisions for individuals. For such applications, we need to strike a balance between achieving good prediction accuracy and making fair decisions with respect to a sensitive feature (e.g., race or gender), which is difficult in complex real-world scenarios. Existing methods measure the unfairness in such scenarios as {\it unfair causal effects} and constrain its mean to zero. Unfortunately, with these methods, the decisions are not necessarily fair for all individuals because even when the mean unfair effect is zero, unfair effects might be positive for some individuals and negative for others, which is discriminatory for them. To learn a classifier that is fair for all individuals, we define unfairness as the {\it probability of individual unfairness} (PIU) and propose to solve an optimization problem that constrains an upper bound on PIU. We theoretically illustrate why our method achieves individual fairness. Experimental results demonstrate that our method learns an individually fair classifier at a slight cost of prediction accuracy.

In reinforcement learning, an agent attempts to learn high-performing behaviors through interacting with the environment, such behaviors are often quantified in the form of a reward function. However some aspects of behavior, such as ones which are deemed unsafe and are to be avoided, are best captured through constraints. We propose a novel approach called First Order Constrained Optimization in Policy Space (FOCOPS) which maximizes an agent's overall reward while ensuring the agent satisfies a set of cost constraints. Using data generated from the current policy, FOCOPS first finds the optimal update policy by solving a constrained optimization problem in the nonparameterized policy space. FOCOPS then projects the update policy back into the parametric policy space. Our approach provides a guarantee for constraint satisfaction throughout training and is first-order in nature therefore extremely simple to implement. We provide empirical evidence that our algorithm achieves better performance on a set of constrained robotics locomotive tasks compared to current state of the art approaches.

We present a new algorithmic approach for the task of finding a chordal Markov network structure that maximizes a given scoring function. The algorithm is based on branch and bound and integrates dynamic programming for both domain pruning and for obtaining strong bounds for search-space pruning. Empirically, we show that the approach dominates in terms of running times a recent integer programming approach (and thereby also a recent constraint optimization approach) for the problem. Papers published at the Neural Information Processing Systems Conference.

We present an algorithm called Optimistic Linear Programming (OLP) for learning to optimize average reward in an irreducible but otherwise unknown Markov decision process (MDP). OLP uses its experience so far to estimate the MDP. It chooses actions by optimistically maximizing estimated future rewards over a set of next-state transition probabilities that are close to the estimates: a computation that corresponds to solving linear programs. We show that the total expected reward obtained by OLP up to time $T$ is within $C(P)\log T$ of the reward obtained by the optimal policy, where $C(P)$ is an explicit, MDP-dependent constant. OLP is closely related to an algorithm proposed by Burnetas and Katehakis with four key differences: OLP is simpler, it does not require knowledge of the supports of transition probabilities and the proof of the regret bound is simpler, but our regret bound is a constant factor larger than the regret of their algorithm.

The control of high-dimensional, continuous, non-linear systems is a key problem in reinforcement learning and control. Local, trajectory-based methods, using techniques such as Differential Dynamic Programming (DDP) are not directly subject to the curse of dimensionality, but generate only local controllers. In this paper, we introduce Receding Horizon DDP (RH-DDP), an extension to the classic DDP algorithm, which allows us to construct stable and robust controllers based on a library of local-control trajectories. We demonstrate the effectiveness of our approach on a series of high-dimensional control problems using a simulated multi-link swimming robot. These experiments show that our approach effectively circumvents dimensionality issues, and is capable of dealing effectively with problems with (at least) 34 state and 14 action dimensions.

Loopy belief propagation has been employed in a wide variety of applications with great empirical success, but it comes with few theoretical guarantees. In this paper we investigate the use of the max-product form of belief propagation for weighted matching problems on general graphs. We show that max-product converges to the correct answer if the linear programming (LP) relaxation of the weighted matching problem is tight and does not converge if the LP relaxation is loose. This provides an exact characterization of max-product performance and reveals connections to the widely used optimization technique of LP relaxation. In addition, we demonstrate that max-product is effective in solving practical weighted matching problems in a distributed fashion by applying it to the problem of self-organization in sensor networks.

Active learning sequentially selects unlabeled instances to label with the goal of reducing the effort needed to learn a good classifier. Most previous studies in active learning have focused on selecting one unlabeled instance at one time while retraining in each iteration. However, single instance selection systems are unable to exploit a parallelized labeler when one is available. Recently a few batch mode active learning approaches have been proposed that select a set of most informative unlabeled instances in each iteration, guided by some heuristic scores. In this paper, we propose a discriminative batch mode active learning approach that formulates the instance selection task as a continuous optimization problem over auxiliary instance selection variables.

We combine two threads of research on approximate dynamic programming: random sampling of states and using local trajectory optimizers to globally optimize a policy and associated value function. This combination allows us to replace a dense multidimensional grid with a much sparser adaptive sampling of states. Our focus is on finding steady state policies for the deterministic time invariant discrete time control problems with continuous states and actions often found in robotics. In this paper we show that we can now solve problems we couldn't solve previously with regular grid-based approaches. Papers published at the Neural Information Processing Systems Conference.

We consider robust least-squares regression with feature-wise disturbance. We show that this formulation leads to tractable convex optimization problems, and we exhibit a particular uncertainty set for which the robust problem is equivalent to $\ell_1$ regularized regression (Lasso). This provides an interpretation of Lasso from a robust optimization perspective. We generalize this robust formulation to consider more general uncertainty sets, which all lead to tractable convex optimization problems. Therefore, we provide a new methodology for designing regression algorithms, which generalize known formulations.