In the blind deconvolution problem, we observe the convolution of an unknown filter and unknown signal and attempt to reconstruct the filter and signal. The problem seems impossible in general, since there are seemingly many more unknowns than knowns . Nevertheless, this problem arises in many application fields; and empirically, some of these fields have had success using heuristic methods -- even economically very important ones, in wireless communications and oil exploration. Today's fashionable heuristic formulations pose non-convex optimization problems which are then attacked heuristically as well. The fact that blind deconvolution can be solved under some repeatable and naturally-occurring circumstances poses a theoretical puzzle. To bridge the gulf between reported successes and theory's limited understanding, we exhibit a convex optimization problem that -- assuming signal sparsity -- can convert a crude approximation to the true filter into a high-accuracy recovery of the true filter. Our proposed formulation is based on L1 minimization of inverse filter outputs. We give sharp guarantees on performance of the minimizer assuming sparsity of signal, showing that our proposal precisely recovers the true inverse filter, up to shift and rescaling. There is a sparsity/initial accuracy tradeoff: the less accurate the initial approximation, the greater we rely on sparsity to enable exact recovery. To our knowledge this is the first reported tradeoff of this kind. We consider it surprising that this tradeoff is independent of dimension. We also develop finite-$N$ guarantees, for highly accurate reconstruction under $N\geq O(k \log(k) )$ with high probability. We further show stable approximation when the true inverse filter is infinitely long and extend our guarantees to the case where the observations are contaminated by stochastic or adversarial noise.

This paper aims to solve a distributed learning problem under Byzantine attacks. In the underlying distributed system, a number of unknown but malicious workers (termed as Byzantine workers) can send arbitrary messages to the master and bias the learning process, due to data corruptions, computation errors or malicious attacks. Prior work has considered a total variation (TV) norm-penalized approximation formulation to handle the Byzantine attacks, where the TV norm penalty forces the regular workers' local variables to be close, and meanwhile, tolerates the outliers sent by the Byzantine workers. To solve the TV norm-penalized approximation formulation, we propose a Byzantine-robust stochastic alternating direction method of multipliers (ADMM) that fully utilizes the separable problem structure. Theoretically, we prove that the proposed method converges to a bounded neighborhood of the optimal solution at a rate of O(1/k) under mild assumptions, where k is the number of iterations and the size of neighborhood is determined by the number of Byzantine workers. Numerical experiments on the MNIST and COVERTYPE datasets demonstrate the effectiveness of the proposed method to various Byzantine attacks.

We study a stochastic program where the probability distribution of the uncertain problem parameters is unknown and only indirectly observed via finitely many correlated samples generated by an unknown Markov chain with $d$ states. We propose a data-driven distributionally robust optimization model to estimate the problem's objective function and optimal solution. By leveraging results from large deviations theory, we derive statistical guarantees on the quality of these estimators. The underlying worst-case expectation problem is nonconvex and involves $\mathcal O(d^2)$ decision variables. Thus, it cannot be solved efficiently for large $d$. By exploiting the structure of this problem, we devise a customized Frank-Wolfe algorithm with convex direction-finding subproblems of size $\mathcal O(d)$. We prove that this algorithm finds a stationary point efficiently under mild conditions. The efficiency of the method is predicated on a dimensionality reduction enabled by a dual reformulation. Numerical experiments indicate that our approach has better computational and statistical properties than the state-of-the-art methods.

We consider the problem of constrained Markov Decision Process (CMDP) where an agent interacts with a unichain Markov Decision Process. At every interaction, the agent obtains a reward. Further, there are $K$ cost functions. The agent aims to maximize the long-term average reward while simultaneously keeping the $K$ long-term average costs lower than a certain threshold. In this paper, we propose CMDP-PSRL, a posterior sampling based algorithm using which the agent can learn optimal policies to interact with the CMDP. Further, for MDP with $S$ states, $A$ actions, and diameter $D$, we prove that following CMDP-PSRL algorithm, the agent can bound the regret of not accumulating rewards from optimal policy by $\Tilde{O}(poly(DSA)\sqrt{T})$. Further, we show that the violations for any of the $K$ constraints is also bounded by $\Tilde{O}(poly(DSA)\sqrt{T})$. To the best of our knowledge, this is the first work which obtains a $\Tilde{O}(\sqrt{T})$ regret bounds for ergodic MDPs with long-term average constraints.

Hyperparameter optimization in machine learning (ML) deals with the problem of empirically learning an optimal algorithm configuration from data, usually formulated as a black-box optimization problem. In this work, we propose a zero-shot method to meta-learn symbolic default hyperparameter configurations that are expressed in terms of the properties of the dataset. This enables a much faster, but still data-dependent, configuration of the ML algorithm, compared to standard hyperparameter optimization approaches. In the past, symbolic and static default values have usually been obtained as hand-crafted heuristics. We propose an approach of learning such symbolic configurations as formulas of dataset properties from a large set of prior evaluations on multiple datasets by optimizing over a grammar of expressions using an evolutionary algorithm. We evaluate our method on surrogate empirical performance models as well as on real data across 6 ML algorithms on more than 100 datasets and demonstrate that our method indeed finds viable symbolic defaults.

The difficulty in specifying rewards for many real-world problems has led to an increased focus on learning rewards from human feedback, such as demonstrations. However, there are often many different reward functions that explain the human feedback, leaving agents with uncertainty over what the true reward function is. While most policy optimization approaches handle this uncertainty by optimizing for expected performance, many applications demand risk-averse behavior. We derive a novel policy gradient-style robust optimization approach, PG-BROIL, that optimizes a soft-robust objective that balances expected performance and risk. To the best of our knowledge, PG-BROIL is the first policy optimization algorithm robust to a distribution of reward hypotheses which can scale to continuous MDPs. Results suggest that PG-BROIL can produce a family of behaviors ranging from risk-neutral to risk-averse and outperforms state-of-the-art imitation learning algorithms when learning from ambiguous demonstrations by hedging against uncertainty, rather than seeking to uniquely identify the demonstrator's reward function.

Bayesian optimization (BO) is a popular method for optimizing expensive-to-evaluate black-box functions. BO budgets are typically given in iterations, which implicitly assumes each evaluation has the same cost. In fact, in many BO applications, evaluation costs vary significantly in different regions of the search space. In hyperparameter optimization, the time spent on neural network training increases with layer size; in clinical trials, the monetary cost of drug compounds vary; and in optimal control, control actions have differing complexities. Cost-constrained BO measures convergence with alternative cost metrics such as time, money, or energy, for which the sample efficiency of standard BO methods is ill-suited. For cost-constrained BO, cost efficiency is far more important than sample efficiency. In this paper, we formulate cost-constrained BO as a constrained Markov decision process (CMDP), and develop an efficient rollout approximation to the optimal CMDP policy that takes both the cost and future iterations into account. We validate our method on a collection of hyperparameter optimization problems as well as a sensor set selection application.

Min-max saddle point games have recently been intensely studied, due to their wide range of applications, including training Generative Adversarial Networks~(GANs). However, most of the recent efforts for solving them are limited to special regimes such as convex-concave games. Further, it is customarily assumed that the underlying optimization problem is solved either by a single machine or in the case of multiple machines connected in centralized fashion, wherein each one communicates with a central node. The latter approach becomes challenging, when the underlying communications network has low bandwidth. In addition, privacy considerations may dictate that certain nodes can communicate with a subset of other nodes. Hence, it is of interest to develop methods that solve min-max games in a decentralized manner. To that end, we develop a decentralized adaptive momentum (ADAM)-type algorithm for solving min-max optimization problem under the condition that the objective function satisfies a Minty Variational Inequality condition, which is a generalization to convex-concave case. The proposed method overcomes shortcomings of recent non-adaptive gradient-based decentralized algorithms for min-max optimization problems that do not perform well in practice and require careful tuning. In this paper, we obtain non-asymptotic rates of convergence of the proposed algorithm (coined DADAM$^3$) for finding a (stochastic) first-order Nash equilibrium point and subsequently evaluate its performance on training GANs. The extensive empirical evaluation shows that DADAM$^3$ outperforms recently developed methods, including decentralized optimistic stochastic gradient for solving such min-max problems.

The Vehicle Routing Problem (VRP) is the combinatorial optimization problem of designing routes for vehicles to visit customers in such a fashion that a cost function, typically the number of vehicles, or the total travelled distance is minimized. The problem finds applications in industrial scenarios, for example where Automated Guided Vehicles run through the plant to deliver components from the warehouse. This specific problem, henceforth called the Electric Conflict-Free Vehicle Routing Problem (CF-EVRP), involves constraints such as limited operating range of the vehicles, time windows on the delivery to the customers, and limited capacity on the number of vehicles the road segments can accommodate at the same time. Such a complex system results in a large model that cannot easily be solved to optimality in reasonable time. We therefore developed a compositional model that breaks down the problem into smaller and simpler sub-problems and provides sub-optimal, feasible solutions to the original problem. The algorithm exploits the strengths of SMT solvers, which proved in our previous work to be an efficient approach to deal with scheduling problems. Compared to a monolithic model for the CF-EVRP, written in the SMT standard language and solved using a state-of-the-art SMT solver the compositional model was found to be significantly faster.

In this paper, we propose a realistic multiple dynamic pricing approach to demand response in the retail market. First, an adaptive clustering-based customer segmentation framework is proposed to categorize customers into different groups to enable the effective identification of usage patterns. Second, customized demand models with important market constraints which capture the price-demand relationship explicitly, are developed for each group of customers to improve the model accuracy and enable meaningful pricing. Third, the multiple pricing based demand response is formulated as a profit maximization problem subject to realistic market constraints. The overall aim of the proposed scalable and practical method aims to achieve 'right' prices for 'right' customers so as to benefit various stakeholders in the system such as grid operators, customers and retailers. The proposed multiple pricing framework is evaluated via simulations based on real-world datasets.