Cooperative Online Learning: Keeping your Neighbors Updated

We study an asynchronous online learning setting with a network of agents. At each time step, some of the agents are activated, requested to make a prediction, and pay the corresponding loss. The loss function is then revealed to these agents and also to their neighbors in the network. Our results characterize how much knowing the network structure affects the regret as a function of the model of agent activations. When activations are stochastic, the optimal regret (up to constant factors) is shown to be of order $\sqrt{\alpha T}$, where $T$ is the horizon and $\alpha$ is the independence number of the network. We prove that the upper bound is achieved even when agents have no information about the network structure. When activations are adversarial the situation changes dramatically: if agents ignore the network structure, a $\Omega(T)$ lower bound on the regret can be proven, showing that learning is impossible. However, when agents can choose to ignore some of their neighbors based on the knowledge of the network structure, we prove a $O(\sqrt{\overline{\chi} T})$ sublinear regret bound, where $\overline{\chi} \ge \alpha$ is the clique-covering number of the network

Dynamic Metric Learning from Pairwise Comparisons

Recent work in distance metric learning has focused on learning transformations of data that best align with specified pairwise similarity and dissimilarity constraints, often supplied by a human observer. The learned transformations lead to improved retrieval, classification, and clustering algorithms due to the better adapted distance or similarity measures. Here, we address the problem of learning these transformations when the underlying constraint generation process is nonstationary. This nonstationarity can be due to changes in either the ground-truth clustering used to generate constraints or changes in the feature subspaces in which the class structure is apparent. We propose Online Convex Ensemble StrongLy Adaptive Dynamic Learning (OCELAD), a general adaptive, online approach for learning and tracking optimal metrics as they change over time that is highly robust to a variety of nonstationary behaviors in the changing metric. We apply the OCELAD framework to an ensemble of online learners. Specifically, we create a retro-initialized composite objective mirror descent (COMID) ensemble (RICE) consisting of a set of parallel COMID learners with different learning rates, demonstrate RICE-OCELAD on both real and synthetic data sets and show significant performance improvements relative to previously proposed batch and online distance metric learning algorithms.Comment: to appear Allerton 2016. arXiv admin note: substantial text overlap with arXiv:1603.0367

Lipschitz Adaptivity with Multiple Learning Rates in Online Learning

We aim to design adaptive online learning algorithms that take advantage of any special structure that might be present in the learning task at hand, with as little manual tuning by the user as possible. A fundamental obstacle that comes up in the design of such adaptive algorithms is to calibrate a so-called step-size or learning rate hyperparameter depending on variance, gradient norms, etc. A recent technique promises to overcome this difficulty by maintaining multiple learning rates in parallel. This technique has been applied in the MetaGrad algorithm for online convex optimization and the Squint algorithm for prediction with expert advice. However, in both cases the user still has to provide in advance a Lipschitz hyperparameter that bounds the norm of the gradients. Although this hyperparameter is typically not available in advance, tuning it correctly is crucial: if it is set too small, the methods may fail completely; but if it is taken too large, performance deteriorates significantly. In the present work we remove this Lipschitz hyperparameter by designing new versions of MetaGrad and Squint that adapt to its optimal value automatically. We achieve this by dynamically updating the set of active learning rates. For MetaGrad, we further improve the computational efficiency of handling constraints on the domain of prediction, and we remove the need to specify the number of rounds in advance.Comment: 22 pages. To appear in COLT 201

A Second-order Bound with Excess Losses

We study online aggregation of the predictions of experts, and first show new second-order regret bounds in the standard setting, which are obtained via a version of the Prod algorithm (and also a version of the polynomially weighted average algorithm) with multiple learning rates. These bounds are in terms of excess losses, the differences between the instantaneous losses suffered by the algorithm and the ones of a given expert. We then demonstrate the interest of these bounds in the context of experts that report their confidences as a number in the interval [0,1] using a generic reduction to the standard setting. We conclude by two other applications in the standard setting, which improve the known bounds in case of small excess losses and show a bounded regret against i.i.d. sequences of losses

The learning curve in a competitive industry.

We consider the learning curve in an industry with free entry and exit, and price-taking firms. A unique equilibrium exists if the fixed cost is positive. While equilibrium profits are zero, mature firms earn rents on their learning, and, if costs are convex, no firm can profitably enter after the date the industry begins. Under some cost and demand conditions, however, firms may have to exit the market despite their experience gained earlier. Furthermore identical firms facing the same prices may produce different quantities. The market outcome is always socially efficient, even if dictates that firms exit after learning. Finally, actual and optimal industry concentration does not always increase in the intensity of learning.Learning curve; Industry evolution; Perfect competition;