Fredrik Hellström

Fredrik Hellström, Giuseppe Durisi

We present a new family of information-theoretic generalization bounds, in which the training loss and the population loss are compared through a jointly convex function. This function is upper-bounded in terms of the disintegrated, samplewise, evaluated conditional mutual information (CMI), an information measure that depends on the losses incurred by the selected hypothesis, rather than on the hypothesis itself, as is common in probably approximately correct (PAC)-Bayesian results. We demonstrate the generality of this framework by recovering and extending previously known information-theoretic bounds. Furthermore, using the evaluated CMI, we derive a samplewise, average version of Seeger's PAC-Bayesian bound, where the convex function is the binary KL divergence. In some scenarios, this novel bound results in a tighter characterization of the population loss of deep neural networks than previous bounds. Finally, we derive high-probability versions of some of these average bounds. We demonstrate the unifying nature of the evaluated CMI bounds by using them to recover average and high-probability generalization bounds for multiclass classification with finite Natarajan dimension.

Fredrik Hellström, Giuseppe Durisi

Recent work has established that the conditional mutual information (CMI) framework of Steinke and Zakynthinou (2020) is expressive enough to capture generalization guarantees in terms of algorithmic stability, VC dimension, and related complexity measures for conventional learning (Harutyunyan et al., 2021, Haghifam et al., 2021). Hence, it provides a unified method for establishing generalization bounds. In meta learning, there has so far been a divide between information-theoretic results and results from classical learning theory. In this work, we take a first step toward bridging this divide. Specifically, we present novel generalization bounds for meta learning in terms of the evaluated CMI (e-CMI). To demonstrate the expressiveness of the e-CMI framework, we apply our bounds to a representation learning setting, with $n$ samples from $\hat n$ tasks parameterized by functions of the form $f_i \circ h$. Here, each $f_i \in \mathcal F$ is a task-specific function, and $h \in \mathcal H$ is the shared representation. For this setup, we show that the e-CMI framework yields a bound that scales as $\sqrt{ \mathcal C(\mathcal H)/(n\hat n) + \mathcal C(\mathcal F)/n} $, where $\mathcal C(\cdot)$ denotes a complexity measure of the hypothesis class. This scaling behavior coincides with the one reported in Tripuraneni et al. (2020) using Gaussian complexity.

Rui M. Castro, Fredrik Hellström, Tim van Erven

We consider online prediction of a binary sequence with expert advice. For this setting, we devise label-efficient forecasting algorithms, which use a selective sampling scheme that enables collecting much fewer labels than standard procedures, while still retaining optimal worst-case regret guarantees. These algorithms are based on exponentially weighted forecasters, suitable for settings with and without a perfect expert. For a scenario where one expert is strictly better than the others in expectation, we show that the label complexity of the label-efficient forecaster scales roughly as the square root of the number of rounds. Finally, we present numerical experiments empirically showing that the normalized regret of the label-efficient forecaster can asymptotically match known minimax rates for pool-based active learning, suggesting it can optimally adapt to benign settings.

Fredrik Hellström, Benjamin Guedj

We derive generic information-theoretic and PAC-Bayesian generalization bounds involving an arbitrary convex comparator function, which measures the discrepancy between the training and population loss. The bounds hold under the assumption that the cumulant-generating function (CGF) of the comparator is upper-bounded by the corresponding CGF within a family of bounding distributions. We show that the tightest possible bound is obtained with the comparator being the convex conjugate of the CGF of the bounding distribution, also known as the Cramér function. This conclusion applies more broadly to generalization bounds with a similar structure. This confirms the near-optimality of known bounds for bounded and sub-Gaussian losses and leads to novel bounds under other bounding distributions.

Fredrik Hellström, Giuseppe Durisi

We present a framework to derive bounds on the test loss of randomized learning algorithms for the case of bounded loss functions. Drawing from Steinke & Zakynthinou (2020), this framework leads to bounds that depend on the conditional information density between the the output hypothesis and the choice of the training set, given a larger set of data samples from which the training set is formed. Furthermore, the bounds pertain to the average test loss as well as to its tail probability, both for the PAC-Bayesian and the single-draw settings. If the conditional information density is bounded uniformly in the size $n$ of the training set, our bounds decay as $1/n$. This is in contrast with the tail bounds involving conditional information measures available in the literature, which have a less benign $1/\sqrt{n}$ dependence. We demonstrate the usefulness of our tail bounds by showing that they lead to nonvacuous estimates of the test loss achievable with some neural network architectures trained on MNIST and Fashion-MNIST.