Oracle Efficient Online Multicalibration and Omniprediction

A recent line of work has shown a surprising connection between multicalibration, a multi-group fairness notion, and omniprediction, a learning paradigm that provides simultaneous loss minimization guarantees for a large family of loss functions. Prior work studies omniprediction in the batch setting. We initiate the study of omniprediction in the online adversarial setting. Although there exist algorithms for obtaining notions of multicalibration in the online adversarial setting, unlike batch algorithms, they work only for small finite classes of benchmark functions $F$, because they require enumerating every function $f \in F$ at every round. In contrast, omniprediction is most interesting for learning theoretic hypothesis classes $F$, which are generally continuously large. We develop a new online multicalibration algorithm that is well defined for infinite benchmark classes $F$, and is oracle efficient (i.e. for any class $F$, the algorithm has the form of an efficient reduction to a no-regret learning algorithm for $F$). The result is the first efficient online omnipredictor -- an oracle efficient prediction algorithm that can be used to simultaneously obtain no regret guarantees to all Lipschitz convex loss functions. For the class $F$ of linear functions, we show how to make our algorithm efficient in the worst case. Also, we show upper and lower bounds on the extent to which our rates can be improved: our oracle efficient algorithm actually promises a stronger guarantee called swap-omniprediction, and we prove a lower bound showing that obtaining $O(\sqrt{T})$ bounds for swap-omniprediction is impossible in the online setting. On the other hand, we give a (non-oracle efficient) algorithm which can obtain the optimal $O(\sqrt{T})$ omniprediction bounds without going through multicalibration, giving an information theoretic separation between these two solution concepts.

Memory-Sample Lower Bounds for Learning Parity with Noise

In this work, we show, for the well-studied problem of learning parity under noise, where a learner tries to learn $x=(x_1,\ldots,x_n) \in \{0,1\}^n$ from a stream of random linear equations over $\mathrm{F}_2$ that are correct with probability $\frac{1}{2}+\varepsilon$ and flipped with probability $\frac{1}{2}-\varepsilon$, that any learning algorithm requires either a memory of size $\Omega(n^2/\varepsilon)$ or an exponential number of samples. In fact, we study memory-sample lower bounds for a large class of learning problems, as characterized by [GRT'18], when the samples are noisy. A matrix $M: A \times X \rightarrow \{-1,1\}$ corresponds to the following learning problem with error parameter $\varepsilon$: an unknown element $x \in X$ is chosen uniformly at random. A learner tries to learn $x$ from a stream of samples, $(a_1, b_1), (a_2, b_2) \ldots$, where for every $i$, $a_i \in A$ is chosen uniformly at random and $b_i = M(a_i,x)$ with probability $1/2+\varepsilon$ and $b_i = -M(a_i,x)$ with probability $1/2-\varepsilon$ ($0<\varepsilon< \frac{1}{2}$). Assume that $k,\ell, r$ are such that any submatrix of $M$ of at least $2^{-k} \cdot |A|$ rows and at least $2^{-\ell} \cdot |X|$ columns, has a bias of at most $2^{-r}$. We show that any learning algorithm for the learning problem corresponding to $M$, with error, requires either a memory of size at least $\Omega\left(\frac{k \cdot \ell}{\varepsilon} \right)$, or at least $2^{\Omega(r)}$ samples. In particular, this shows that for a large class of learning problems, same as those in [GRT'18], any learning algorithm requires either a memory of size at least $\Omega\left(\frac{(\log |X|) \cdot (\log |A|)}{\varepsilon}\right)$ or an exponential number of noisy samples. Our proof is based on adapting the arguments in [Raz'17,GRT'18] to the noisy case.

The Role of Randomness and Noise in Strategic Classification

We investigate the problem of designing optimal classifiers in the strategic classification setting, where the classification is part of a game in which players can modify their features to attain a favorable classification outcome (while incurring some cost). Previously, the problem has been considered from a learning-theoretic perspective and from the algorithmic fairness perspective. Our main contributions include 1. Showing that if the objective is to maximize the efficiency of the classification process (defined as the accuracy of the outcome minus the sunk cost of the qualified players manipulating their features to gain a better outcome), then using randomized classifiers (that is, ones where the probability of a given feature vector to be accepted by the classifier is strictly between 0 and 1) is necessary. 2. Showing that in many natural cases, the imposed optimal solution (in terms of efficiency) has the structure where players never change their feature vectors (the randomized classifier is structured in a way, such that the gain in the probability of being classified as a 1 does not justify the expense of changing one's features). 3. Observing that the randomized classification is not a stable best-response from the classifier's viewpoint, and that the classifier doesn't benefit from randomized classifiers without creating instability in the system. 4. Showing that in some cases, a noisier signal leads to better equilibria outcomes -- improving both accuracy and fairness when more than one subpopulation with different feature adjustment costs are involved. This is interesting from a policy perspective, since it is hard to force institutions to stick to a particular randomized classification strategy (especially in a context of a market with multiple classifiers), but it is possible to alter the information environment to make the feature signals inherently noisier.

Tracking and Improving Information in the Service of Fairness

As algorithmic prediction systems have become widespread, fears that these systems may inadvertently discriminate against members of underrepresented populations have grown. With the goal of understanding fundamental principles that underpin the growing number of approaches to mitigating algorithmic discrimination, we investigate the role of information in fair prediction. A common strategy for decision-making uses a predictor to assign individuals a risk score; then, individuals are selected or rejected on the basis of this score. In this work, we formalize a framework for measuring the information content of predictors. Central to this framework is the notion of a refinement; intuitively, a refinement of a predictor $z$ increases the overall informativeness of the predictions without losing the information already contained in $z$. We show that increasing information content through refinements improves the downstream selection rules across a wide range of fairness measures (e.g. true positive rates, false positive rates, selection rates). In turn, refinements provide a simple but effective tool for reducing disparity in treatment and impact without sacrificing the utility of the predictions. Our results suggest that in many applications, the perceived "cost of fairness" results from an information disparity across populations, and thus, may be avoided with improved information.

Extractor-Based Time-Space Lower Bounds for Learning

A matrix $M: A \times X \rightarrow \{-1,1\}$ corresponds to the following learning problem: An unknown element $x \in X$ is chosen uniformly at random. A learner tries to learn $x$ from a stream of samples, $(a_1, b_1), (a_2, b_2) \ldots$, where for every $i$, $a_i \in A$ is chosen uniformly at random and $b_i = M(a_i,x)$. Assume that $k,\ell, r$ are such that any submatrix of $M$ of at least $2^{-k} \cdot |A|$ rows and at least $2^{-\ell} \cdot |X|$ columns, has a bias of at most $2^{-r}$. We show that any learning algorithm for the learning problem corresponding to $M$ requires either a memory of size at least $\Omega\left(k \cdot \ell \right)$, or at least $2^{\Omega(r)}$ samples. The result holds even if the learner has an exponentially small success probability (of $2^{-\Omega(r)}$). In particular, this shows that for a large class of learning problems, any learning algorithm requires either a memory of size at least $\Omega\left((\log |X|) \cdot (\log |A|)\right)$ or an exponential number of samples, achieving a tight $\Omega\left((\log |X|) \cdot (\log |A|)\right)$ lower bound on the size of the memory, rather than a bound of $\Omega\left(\min\left\{(\log |X|)^2,(\log |A|)^2\right\}\right)$ obtained in previous works [R17,MM17b]. Moreover, our result implies all previous memory-samples lower bounds, as well as a number of new applications. Our proof builds on [R17] that gave a general technique for proving memory-samples lower bounds.