This paper considers online convex optimization over a complicated constraint set, which typically consists of multiple functional constraints and a set constraint. The conventional online projection algorithm (Zinkevich, 2003) can be difficult to implement due to the potentially high computation complexity of the projection operation. In this paper, we relax the functional constraints by allowing them to be violated at each round but still requiring them to be satisfied in the long term. This type of relaxed online convex optimization (with long term constraints) was first considered in Mahdavi et al. (2012). That prior work proposes an algorithm to achieve $O(\sqrt{T})$ regret and $O(T^{3/4})$ constraint violations for general problems and another algorithm to achieve an $O(T^{2/3})$ bound for both regret and constraint violations when the constraint set can be described by a finite number of linear constraints. A recent extension in Jenatton et al. (2016) can achieve $O(T^{\max\{\theta,1-\theta\}})$ regret and $O(T^{1-\theta/2})$ constraint violations where $\theta\in (0,1)$. The current paper proposes a new simple algorithm that yields improved performance in comparison to prior works. The new algorithm achieves an $O(\sqrt{T})$ regret bound with $O(1)$ constraint violations.

This paper considers utility optimal power control for energy-harvesting wireless devices with a finite capacity battery. The distribution information of the underlying wireless environment and harvestable energy is unknown, and only outdated system state information is known at the device controller. This scenario shares similarity with Lyapunov opportunistic optimization and online learning but is different from both. By a novel combination of Zinkevich's online gradient learning technique and the drift-plus-penalty technique from Lyapunov opportunistic optimization, this paper proposes a learning-aided algorithm that achieves utility within O(ϵ) of the optimal, for any desired ϵ > 0, by using a battery with an O(1/ϵ) capacity. The proposed algorithm has low complexity and makes power investment decisions based on system history, without requiring knowledge of the system state or its probability distribution.

This paper considers utility optimal power control for energy harvesting wireless devices with a finite capacity battery. The distribution information of the underlying wireless environment and harvestable energy is unknown and only outdated system state information is known at the device controller. This scenario shares similarity with Lyapunov opportunistic optimization and online learning but is different from both. By a novel combination of Zinkevich's online gradient learning technique and the drift-plus-penalty technique from Lyapunov opportunistic optimization, this paper proposes a learning-aided algorithm that achieves utility within O (ϵ) of the optimal, for any desired ϵ > 0, by using a battery with an 0 (1/ϵ) capacity. The proposed algorithm has low complexity and makes power investment decisions based on system history, without requiring knowledge of the system state or its probability distribution.

We propose a new primal-dual homotopy smoothing algorithm for a linearly constrained convex program, where neither the primal nor the dual function has to be smooth or strongly convex. The best known iteration complexity solving such a non-smooth problem is $\mathcal{O}(\varepsilon^{-1})$. In this paper, we show that by leveraging a local error bound condition on the dual function, the proposed algorithm can achieve a better primal convergence time of $\mathcal{O}\l(\varepsilon^{-2/(2+\beta)}\log_2(\varepsilon^{-1})\r)$, where $\beta\in(0,1]$ is a local error bound parameter. As an example application, we show that the distributed geometric median problem, which can be formulated as a constrained convex program, has its dual function non-smooth but satisfying the aforementioned local error bound condition with $\beta=1/2$, therefore enjoying a convergence time of $\mathcal{O}\l(\varepsilon^{-4/5}\log_2(\varepsilon^{-1})\r)$. This result improves upon the $\mathcal{O}(\varepsilon^{-1})$ convergence time bound achieved by existing distributed optimization algorithms. Simulation experiments also demonstrate the performance of our proposed algorithm.

The backpressure algorithm has been widely used as a distributed solution to the problem of joint rate control and routing in multi-hop data networks. By controlling a parameter V in the algorithm, the backpressure algorithm can achieve an arbitrarily small utility optimality gap. However, this in turn brings in a large queue length at each node and hence causes large network delay. This phenomenon is known as the fundamental utility-delay tradeoff. The best known utility-delay tradeoff for general networks is [O(1/V), O(V)] and is attained by a backpressure algorithm based on a drift-pluspenalty technique. This may suggest that to achieve an arbitrarily small utility optimality gap, the existing backpressure algorithms necessarily yield an arbitrarily large queue length. However, this paper proposes a new backpressure algorithm that has a vanishing utility optimality gap, so utility converges to exact optimality as the algorithm keeps running, while queue lengths are bounded throughout by a finite constant. The technique uses backpressure and drift concepts with a new method for convex programming.

This paper considers online convex optimization (OCO) with stochastic constraints, which generalizes Zinkevich's OCO over a known simple fixed set by introducing multiple stochastic functional constraints that are i.i.d. generated at each round and are disclosed to the decision maker only after the decision is made. This formulation arises naturally when decisions are restricted by stochastic environments or deterministic environments with noisy observations. It also includes many important problems as special case, such as OCO with long term constraints, stochastic constrained convex optimization, and deterministic constrained convex optimization. To solve this problem, this paper proposes a new algorithm that achieves $O(\sqrt{T})$ expected regret and constraint violations and $O(\sqrt{T}\log(T))$ high probability regret and constraint violations. Experiments on a real-world data center scheduling problem further verify the performance of the new algorithm.

This paper considers dynamic transmit covariance design in point-to-point multiple-input multiple-output fading systems with unknown channel state distributions and inaccurate channel state information subject to both long-term and shortterm power constraints. First, the case of instantaneous but possibly inaccurate channel state information at the transmitter (CSIT) is treated. By extending the drift-plus-penalty technique, a dynamic transmit covariance policy is developed and is shown to approach optimality with an O(δ) gap, where δ is the inaccuracy measure of CSIT, regardless of the channel state distribution and without requiring knowledge of this distribution. Next, the case of delayed and inaccurate channel state information is considered. The optimal transmit covariance solution that maximizes the ergodic capacity is fundamentally different in this case, and a different online algorithm based on convex projections is developed. The proposed algorithm for this delayed-CSIT case also has an O(δ) optimality gap, where δ is again the inaccuracy measure of CSIT.

This paper considers dynamic power allocation in MIMO fading systems with unknown channel state distributions. First, the ideal case of perfect instantaneous channel state information at the transmitter (CSIT) is treated. Using the drift-plus-penalty method, a dynamic power allocation policy is developed and shown to approach optimality, regardless of the channel state distribution and without requiring knowledge of this distribution. Next, the case of delayed and quantized channel state information is considered. Optimal utility is fundamentally different in this case, and a different online algorithm is developed that is based on convex projections. The proposed algorithm for this delayed-CSIT case is shown to have an O (δ) optimality gap, where δ is the quantization error of CSIT.

This paper considers a base station that delivers packets to multiple receivers through a sequence of coded transmissions. All receivers overhear the same transmissions. Each receiver may already have some of the packets as side information, and requests another subset of the packets. This problem is known as the index coding problem and can be represented by a bipartite digraph. An integer linear program is developed that provides a lower bound on the minimum number of transmissions required for any coding algorithm. Conversely, its linear programming relaxation is shown to provide an upper bound that is achievable by a simple form of vector linear coding. Thus, the information theoretic optimum is bounded by the integrality gap between the integer program and its linear relaxation. In the special case, when the digraph has a planar structure, the integrality gap is shown to be zero, so that exact optimality is achieved. Finally, for nonplanar problems, an enhanced integer program is constructed that provides a smaller integrality gap. The dual of this problem corresponds to a more sophisticated partial clique coding strategy that time-shares between maximum distance separable codes. This paper illuminates the relationship between index coding, duality, and integrality gaps between integer programs and their linear relaxations.