Mengdi Wang We study constrained nested stochastic optimization problems in which the objective function is a composition of two smooth functions whose exact values and derivatives are not available. We propose a single time-scale stochastic approximation algorithm, which we call the Nested Averaged Stochastic Approximation (NASA), to find an approximate stationary point of the problem. The algorithm … Read more
Consider the problem of approximating the optimal policy of a Markov decision process (MDP) by sampling state transitions. In contrast to existing reinforcement learning methods that are based on successive approximations to the nonlinear Bellman equation, we propose a Primal-Dual π Learning method in light of the linear duality between the value and policy. The … Read more
We study the computational complexity of the infinite-horizon discounted-reward Markov Decision Problem (MDP) with a finite state space $\cS$ and a finite action space $\cA$. We show that any randomized algorithm needs a running time at least $\Omega(\carS^2\carA)$ to compute an $\epsilon$-optimal policy with high probability. We consider two variants of the MDP where the … Read more
We propose a randomized linear programming algorithm for approximating the optimal policy of the discounted Markov decision problem. By leveraging the value-policy duality, the algorithm adaptively samples state transitions and makes exponentiated primal-dual updates. We show that it finds an ε-optimal policy using nearly-linear running time in the worst case. For Markov decision processes that … Read more
We study the online estimation of the optimal policy of a Markov decision process (MDP). We propose a class of Stochastic Primal-Dual (SPD) methods which exploit the inherent minimax duality of Bellman equations. The SPD methods update a few coordinates of the value and policy estimates as a new state transition is observed. These methods … Read more
In this paper, we consider sparse optimization problems with L0 norm penalty or constraint. We prove that it is strongly NP-hard to find an approximate optimal solution within certain error bound, unless P = NP. This provides a lower bound for the approximation error of any deterministic polynomial-time algorithm. Applying the complexity result to sparse … Read more
We consider the problem of estimating high dimensional spatial graphical models with a total cardinality constraint (i.e., the l0-constraint). Though this problem is highly nonconvex, we show that its primal-dual gap diminishes linearly with the dimensionality and provide a convex geometry justification of this ‘blessing of massive scale’ phenomenon. Motivated by this result, we propose … Read more
Consider convex optimization problems subject to a large number of constraints. We focus on stochastic problems in which the objective takes the form of expected values and the feasible set is the intersection of a large number of convex sets. We propose a class of algorithms that perform both stochastic gradient descent and random feasibility … Read more
We focus on a class of nonconvex cooperative optimization problems that involve multiple participants. We study the duality framework and provide geometric and analytic character- izations of the duality gap. The dual problem is related to a market setting in which each participant pursuits self interests at a given price of common goods. The duality … Read more
Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., problems of the form $\min_x \E_v\[f_v\big(\E_w [g_w(x)]\big) \]$. In order to solve this stochastic composition problem, we propose a class … Read more