Suppose we have many copies of an unknown n-qubit state $\rho$. We measure some copies of $\rho$ using a known two-outcome measurement E_1, then other copies using a measurement E_2, and so on. At each stage t, we generate a current hypothesis $\omega_t$ about the state $\rho$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|\trace(E_i \omega_t) - \trace(E_i\rho)|$, the error in our prediction for the next measurement, is at least $eps$ at most $O(n / eps^2) $\ times. Even in the non-realizable setting---where there could be arbitrary noise in the measurement outcomes---we show how to output hypothesis states that incur at most $O(\sqrt {Tn}) $ excess loss over the best possible state on the first $T$ measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results---using convex optimization, quantum postselection, and sequential fat-shattering dimension---which have different advantages in terms of parameters and portability.

In this paper, we study online convex optimization in dynamic environments, and aim to bound the dynamic regret with respect to any sequence of comparators. Existing work have shown that online gradient descent enjoys an $O(\sqrt{T}(1+P_T))$ dynamic regret, where $T$ is the number of iterations and $P_T$ is the path-length of the comparator sequence. However, this result is unsatisfactory, as there exists a large gap from the $\Omega(\sqrt{T(1+P_T)})$ lower bound established in our paper. To address this limitation, we develop a novel online method, namely adaptive learning for dynamic environment (Ader), which achieves an optimal $O(\sqrt{T(1+P_T)})$ dynamic regret. The basic idea is to maintain a set of experts, each attaining an optimal dynamic regret for a specific path-length, and combines them with an expert-tracking algorithm. Furthermore, we propose an improved Ader based on the surrogate loss, and in this way the number of gradient evaluations per round is reduced from $O(\log T)$ to $1$. Finally, we extend Ader to the setting that a sequence of dynamical models is available to characterize the comparators.

Suppose we have many copies of an unknown n-qubit state $\rho$. We measure some copies of $\rho$ using a known two-outcome measurement E_1, then other copies using a measurement E_2, and so on. At each stage t, we generate a current hypothesis $\omega_t$ about the state $\rho$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|\trace(E_i \omega_t) - \trace(E_i\rho)|$, the error in our prediction for the next measurement, is at least $eps$ at most $O(n / eps^2) $\ times. Even in the non-realizable setting---where there could be arbitrary noise in the measurement outcomes---we show how to output hypothesis states that incur at most $O(\sqrt {Tn}) $ excess loss over the best possible state on the first $T$ measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results---using convex optimization, quantum postselection, and sequential fat-shattering dimension---which have different advantages in terms of parameters and portability.

Implicit feedback, such as user clicks, although abundant in online information service systems, does not provide substantial evidence on users' evaluation of system's output. Without proper modeling, such incomplete supervision inevitably misleads model estimation, especially in a bandit learning setting where the feedback is acquired on the fly. In this work, we perform contextual bandit learning with implicit feedback by modeling the feedback as a composition of user result examination and relevance judgment. Since users' examination behavior is unobserved, we introduce latent variables to model it. We perform Thompson sampling on top of variational Bayesian inference for arm selection and model update. Our upper regret bound analysis of the proposed algorithm proves its feasibility of learning from implicit feedback in a bandit setting; and extensive empirical evaluations on click logs collected from a major MOOC platform further demonstrate its learning effectiveness in practice.

We introduce the community exploration problem that has various real-world applications such as online advertising. In the problem, an explorer allocates limited budget to explore communities so as to maximize the number of members he could meet. We provide a systematic study of the community exploration problem, from offline optimization to online learning. For the offline setting where the sizes of communities are known, we prove that the greedy methods for both of non-adaptive exploration and adaptive exploration are optimal. For the online setting where the sizes of communities are not known and need to be learned from the multi-round explorations, we propose an ``upper confidence'' like algorithm that achieves the logarithmic regret bounds. By combining the feedback from different rounds, we can achieve a constant regret bound.

We consider the problem of online learning in the linear contextual bandits setting, but in which there are also strong individual fairness constraints governed by an unknown similarity metric. These constraints demand that we select similar actions or individuals with approximately equal probability DHPRZ12, which may be at odds with optimizing reward, thus modeling settings where profit and social policy are in tension. We assume we learn about an unknown Mahalanobis similarity metric from only weak feedback that identifies fairness violations, but does not quantify their extent. This is intended to represent the interventions of a regulator who "knows unfairness when he sees it" but nevertheless cannot enunciate a quantitative fairness metric over individuals. Our main result is an algorithm in the adversarial context setting that has a number of fairness violations that depends only logarithmically on T, while obtaining an optimal O(sqrt(T)) regret bound to the best fair policy.

In this paper, we study online convex optimization in dynamic environments, and aim to bound the dynamic regret with respect to any sequence of comparators. Existing work have shown that online gradient descent enjoys an $O(\sqrt{T}(1+P_T))$ dynamic regret, where $T$ is the number of iterations and $P_T$ is the path-length of the comparator sequence. However, this result is unsatisfactory, as there exists a large gap from the $\Omega(\sqrt{T(1+P_T)})$ lower bound established in our paper. To address this limitation, we develop a novel online method, namely adaptive learning for dynamic environment (Ader), which achieves an optimal $O(\sqrt{T(1+P_T)})$ dynamic regret. The basic idea is to maintain a set of experts, each attaining an optimal dynamic regret for a specific path-length, and combines them with an expert-tracking algorithm. Furthermore, we propose an improved Ader based on the surrogate loss, and in this way the number of gradient evaluations per round is reduced from $O(\log T)$ to $1$. Finally, we extend Ader to the setting that a sequence of dynamical models is available to characterize the comparators.

Inverse optimization is a powerful paradigm for learning preferences and restrictions that explain the behavior of a decision maker, based on a set of external signal and the corresponding decision pairs. However, most inverse optimization algorithms are designed specifically in batch setting, where all the data is available in advance. As a consequence, there has been rare use of these methods in an online setting suitable for real-time applications. In this paper, we propose a general framework for inverse optimization through online learning. Specifically, we develop an online learning algorithm that uses an implicit update rule which can handle noisy data. Moreover, under additional regularity assumptions in terms of the data and the model, we prove that our algorithm converges at a rate of $\mathcal{O}(1/\sqrt{T})$ and is statistically consistent. In our experiments, we show the online learning approach can learn the parameters with great accuracy and is very robust to noises, and achieves a dramatic improvement in computational efficacy over the batch learning approach.

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Science fiction visions of the future show us AI built to replicate our way of thinking -- but what if we modeled it instead on the other kinds of intelligence found in nature? Robotics engineer Radhika Nagpal studies the collective intelligence displayed by insects and fish schools, seeking to understand their rules of engagement. In a visionary talk, she presents her work creating artificial collective power and previews a future where swarms of robots work together to build flood barriers, pollinate crops, monitor coral reefs and form constellations of satellites. Radhika Nagpal Taking cues from bottom-up biological networks like those of social insects, Radhika Nagpal helped design an unprecedented "swarm" of ant-like robots. Why you should listen With a swarm of 1,024 robots inspired by the design of ant colonies, Radhika Nagpal and her colleagues at Harvard's SSR research group have redefined expectations for self-organizing robotic systems.