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### Online convex optimization for cumulative constraints

We propose the algorithms for online convex optimization which lead to cumulative squared constraint violations of the form $\sum\limits_{t=1}^T\big([g(x_t)]_+\big)^2=O(T^{1-\beta})$, where $\beta\in(0,1)$. Previous literature has focused on long-term constraints of the form $\sum\limits_{t=1}^Tg(x_t)$. There, strictly feasible solutions can cancel out the effects of violated constraints. In contrast, the new form heavily penalizes large constraint violations and cancellation effects cannot occur. Furthermore, useful bounds on the single step constraint violation $[g(x_t)]_+$ are derived. For convex objectives, our regret bounds generalize existing bounds, and for strongly convex objectives we give improved regret bounds. In numerical experiments, we show that our algorithm closely follows the constraint boundary leading to low cumulative violation.

### Online Convex Optimization in Changing Environments and its Application to Resource Allocation

In the era of the big data, we create and collect lots of data from all different kinds of sources: the Internet, the sensors, the consumer market, and so on. Many of the data are coming sequentially, and would like to be processed and understood quickly. One classic way of analyzing data is based on batch processing, in which the data is stored and analyzed in an offline fashion. However, when the volume of the data is too large, it is much more difficult and time-consuming to do batch processing than sequential processing. What's more, sequential data is usually changing dynamically, and needs to be understood on-the-fly in order to capture the changes. Online Convex Optimization (OCO) is a popular framework that matches the above sequential data processing requirement. Applications using OCO include online routing, online auctions, online classification and regression, as well as online resource allocation. Due to the general applicability of OCO to the sequential data and the rigorous theoretical guarantee, it has attracted lots of researchers to develop useful algorithms to fulfill different needs. In this thesis, we show our contributions to OCO's development by designing algorithms to adapt to changing environments.

### Distributed Online Convex Optimization with Time-Varying Coupled Inequality Constraints

This paper considers distributed online optimization with time-varying coupled inequality constraints. The global objective function is composed of local convex cost and regularization functions and the coupled constraint function is the sum of local convex constraint functions. A distributed online primal-dual dynamic mirror descent algorithm is proposed to solve this problem, where the local cost, regularization, and constraint functions are held privately and revealed only after each time slot. We first derive regret and cumulative constraint violation bounds for the algorithm and show how they depend on the stepsize sequences, the accumulated dynamic variation of the comparator sequence, the number of agents, and the network connectivity. As a result, under some natural decreasing stepsize sequences, we prove that the algorithm achieves sublinear dynamic regret and cumulative constraint violation if the accumulated dynamic variation of the optimal sequence also grows sublinearly. We also prove that the algorithm achieves sublinear static regret and cumulative constraint violation under mild conditions. In addition, smaller bounds on the static regret are achieved when the objective functions are strongly convex. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

### Adaptive Algorithms for Online Convex Optimization with Long-term Constraints

We present an adaptive online gradient descent algorithm to solve online convex optimization problems with long-term constraints , which are constraints that need to be satisfied when accumulated over a finite number of rounds T , but can be violated in intermediate rounds. For some user-defined trade-off parameter $\beta$ $\in$ (0, 1), the proposed algorithm achieves cumulative regret bounds of O(T^max{$\beta$,1--$\beta$}) and O(T^(1--$\beta$/2)) for the loss and the constraint violations respectively. Our results hold for convex losses and can handle arbitrary convex constraints without requiring knowledge of the number of rounds in advance. Our contributions improve over the best known cumulative regret bounds by Mahdavi, et al. (2012) that are respectively O(T^1/2) and O(T^3/4) for general convex domains, and respectively O(T^2/3) and O(T^2/3) when further restricting to polyhedral domains. We supplement the analysis with experiments validating the performance of our algorithm in practice.

### Online Convex Optimization with Stochastic Constraints

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.