We propose a new way of interpreting active learning methods by analyzing the information lost' upon sampling a random variable. We use some recent analytical developments of these losses to formally prove that facility location methods reduce these losses under mild assumptions, and to derive a new data dependent bound on information losses that can be used to evaluate other active learning methods. We show that this new bound is extremely tight to experiment, and further show that the bound has a decent predictive power for classification accuracy.
The second @cite_35 proves that under a Lipschitz continuity condition, empirical risks can be bounded in a way dependent on the maximum distance from any unlabelled point to a labelled one. Thus good data points can be chosen by ensuring that this maximum distance is small. In our work, we will prove that a slight variation of this method reduces information losses - giving the method even stronger theoretical backing.
@cite_35
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We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate. We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks with respect to their inputs. To this end, we provide a simple technique for computing an upper bound to the Lipschitz constant of a feed forward neural network composed of commonly used layer types and demonstrate inaccuracies in previous work on this topic. Our technique is then used to formulate training a neural network with a bounded Lipschitz constant as a constrained optimisation problem that can be solved using projected stochastic gradient methods. Our evaluation study shows that, in isolation, our method performs comparatively to state-of-the-art regularisation techniques. Moreover, when combined with existing approaches to regularising neural networks the performance gains are cumulative. We also provide evidence that the hyperparameters are intuitive to tune and demonstrate how the choice of norm for computing the Lipschitz constant impacts the resulting model. We prove several facts concerning Lipschitz percolation, including the following. The critical probability p_L for the existence of an open Lipschitz surface in site percolation on Z^d with d 2 satisfies the improved bound p_L 1-1 [8(d-1)]. Whenever p > p_L, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. The lowest surface is dominated stochastically by the boundary of a union of certain independent, identically distributed random subsets of Z^d. As a consequence, for p sufficiently close to 1, the connected regions of Z^ d-1 above which the surface has height 2 or more exhibit stretched-exponential tail behaviour. Convex empirical risk minimization is a basic tool in machine learning and statistics. We provide new algorithms and matching lower bounds for differentially private convex empirical risk minimization assuming only that each data point's contribution to the loss function is Lipschitz and that the domain of optimization is bounded. We provide a separate set of algorithms and matching lower bounds for the setting in which the loss functions are known to also be strongly convex.Our algorithms run in polynomial time, and in some cases even match the optimal non-private running time (as measured by oracle complexity). We give separate algorithms (and lower bounds) for (aepsi;, 0)- and (aepsi;,a#x03B4;)-differential privacy, perhaps surprisingly, the techniques used for designing optimal algorithms in the two cases are completely different. Our lower bounds apply even to very simple, smooth function families, such as linear and quadratic functions. This implies that algorithms from previous work can be used to obtain optimal error rates, under the additional assumption that the contributions of each data point to the loss function is smooth. We show that simple approaches to smoothing arbitrary loss functions (in order to apply previous techniques) do not yield optimal error rates. In particular, optimal algorithms were not previously known for problems such as training support vector machines and the high-dimensional median.