Generalized Additive Models (GAMs) are commonly considered *interpretable* within the ML community, as their structure makes the relationship between inputs and outputs relatively understandable. Therefore, it may seem natural to hypothesize that obtaining meaningful explanations for GAMs could be performed efficiently and would not be computationally infeasible. In this work, we challenge this hypothesis by analyzing the *computational complexity* of generating different explanations for various forms of GAMs across multiple contexts. Our analysis reveals a surprisingly diverse landscape of both positive and negative complexity outcomes. Particularly, under standard complexity assumptions such as P$\neq$NP, we establish several key findings: (1) in stark contrast to many other common ML models, the complexity of generating explanations for GAMs is heavily influenced by the structure of the input space; (2) the complexity of explaining GAMs varies significantly with the types of component models used - but interestingly, these differences only emerge under specific input domain settings; (3) significant complexity distinctions appear for obtaining explanations in regression tasks versus classification tasks in GAMs; and (4) expressing complex models like neural networks additively (e.g., as neural additive models) can make them easier to explain, though interestingly, this benefit appears only for certain explanation methods and input domains. Collectively, these results shed light on the feasibility of computing diverse explanations for GAMs, offering a rigorous theoretical picture of the conditions under which such computations are possible or provably hard.

Principal-agent problems arise when one party acts on behalf of another, leading to conflicts of interest. The economic literature has extensively studied principal-agent problems, and recent work has extended this to more complex scenarios such as Markov Decision Processes (MDPs). In this paper, we further explore this line of research by investigating how reward shaping under budget constraints can improve the principal's utility. We study a two-player Stackelberg game where the principal and the agent have different reward functions, and the agent chooses an MDP policy for both players. The principal offers an additional reward to the agent, and the agent picks their policy selfishly to maximize their reward, which is the sum of the original and the offered reward. Our results establish the NP-hardness of the problem and offer polynomial approximation algorithms for two classes of instances: Stochastic trees and deterministic decision processes with a finite horizon.

Simple, sufficient explanations furnished by short decision lists can be useful for guiding stakeholder actions. Unfortunately, this transparency can come at the expense of the higher accuracy enjoyed by black box methods, like deep nets. To date, practitioners typically either (i) insist on the simpler model, forsaking accuracy; or (ii) insist on maximizing accuracy, settling for post-hoc explanations of dubious faithfulness. In this paper, we propose a hybrid partially interpretable model that represents a compromise between the two extremes. In our setup, each input is first processed by a decision list that can either execute a decision or abstain, handing off authority to the opaque model. The key to optimizing the decision list is to optimally trade off the accuracy of the composite system against coverage (the fraction of the population that receives explanations). We contribute a new principled algorithm for constructing partially interpretable decision lists, providing theoretical guarantees addressing both interpretability and accuracy. As an instance of our result, we prove that when the optimal decision list has length $k$, coverage $c$, and $b$ mistakes, our algorithm will generate a decision list that has length no greater than $4k$, coverage at least $c/2$, and makes at most $4b$ mistakes. Finally, we empirically validate the effectiveness of the new model.

We consider $k$-means clustering in an online setting where each new data point is assigned to its closest cluster center and incurs a loss equal to the squared distance to that center, after which the algorithm is allowed to update its centers. The goal over a data stream $X$ is to achieve a total loss that is not too much larger than $L(X, OPT_k)$, the best possible loss using $k$ fixed centers in hindsight. We start by introducing a data parameter, $\Lambda(X)$, such that for any online algorithm that maintains $O(k \, \text{poly}(\log n))$ centers after seeing $n$ points, there exists a data stream $X$ for which a loss of $\Omega(\Lambda(X))$ is inevitable. Next, we give a randomized algorithm that achieves online loss $O(\Lambda(X) + L(X, OPT_k))$, while taking $O(k \, \text{poly}(\log n))$ centers and additional memory. It has an update time of $O(k \, \text{poly}(\log n))$ and is the first algorithm to achieve polynomial space and time complexity in the online setting. We note that our results have implications to the related streaming setting, where one final clustering is outputted, and the no-substitution setting, where center selections are permanent. We show a general reduction between the no-substitution cost of a blackbox algorithm and its online cost. Finally, we translate our algorithm to the no-substitution setting and streaming settings, and it competes with and can outperform existing work in the areas.

Given a policy of a Markov Decision Process, we define a SafeZone as a subset of states, such that most of the policy's trajectories are confined to this subset. The quality of a SafeZone is parameterized by the number of states and the escape probability, i.e., the probability that a random trajectory will leave the subset. SafeZones are especially interesting when they have a small number of states and low escape probability. We study the complexity of finding optimal SafeZones, and show that in general, the problem is computationally hard. For this reason, we concentrate on finding approximate SafeZones. Our main result is a bi-criteria approximation learning algorithm with a factor of almost $2$ approximation for both the escape probability and \newprob size, using a polynomial size sample complexity.

We study the faithfulness of an explanation system to the underlying prediction model. We show that this can be captured by two properties, consistency and sufficiency, and introduce quantitative measures of the extent to which these hold. Interestingly, these measures depend on the test-time data distribution. For a variety of existing explanation systems, such as anchors, we analytically study these quantities. We also provide estimators and sample complexity bounds for empirically determining the faithfulness of black-box explanation systems. Finally, we experimentally validate the new properties and estimators.

We investigate $k$-means clustering in the online no-substitution setting when the input arrives in \emph{arbitrary} order. In this setting, points arrive one after another, and the algorithm is required to instantly decide whether to take the current point as a center before observing the next point. Decisions are irrevocable. The goal is to minimize both the number of centers and the $k$-means cost. Previous works in this setting assume that the input’s order is random, or that the input’s aspect ratio is bounded. It is known that if the order is arbitrary and there is no assumption on the input, then any algorithm must take all points as centers. Moreover, assuming a bounded aspect ratio is too restrictive — it does not include natural input generated from mixture models. We introduce a new complexity measure that quantifies the difficulty of clustering a dataset arriving in arbitrary order. We design a new random algorithm and prove that if applied on data with complexity $d$, the algorithm takes $O(d\log(n) k\log(k))$ centers and is an $O(k^3)$-approximation. We also prove that if the data is sampled from a “natural" distribution, such as a mixture of $k$ Gaussians, then the new complexity measure is equal to $O(k^2\log(n))$. This implies that for data generated from those distributions, our new algorithm takes only $poly(k\log(n))$ centers and is a $poly(k)$-approximation. In terms of negative results, we prove that the number of centers needed to achieve an $\alpha$-approximation is at least $\Omega\left(\frac{d}{k\log(n\alpha)}\right)$.

Offline $k$-means clustering was studied extensively, and algorithms with a constant approximation are available. However, online clustering is still uncharted. New factors come into play: the ordering of the dataset and whether the number of points, $n$, is known in advance or not. Their exact effects are unknown. In this paper we focus on the online setting where the decisions are irreversible: after a point arrives, the algorithm needs to decide whether to take the point as a center or not, and this decision is final. How many centers are needed and sufficient to achieve constant approximation in this setting? We show upper and lower bounds for all the different cases. These bounds are exactly the same up to a constant, thus achieving optimal bounds. For example, for $k$-means cost with constant $k>1$ and random order, $\Theta(\log n)$ centers are enough to achieve a constant approximation, while the mere a priori knowledge of $n$ reduces the number of centers to a constant. These bounds hold for any distance function that obeys a triangle-type inequality.

The explosive growth of easily-accessible unlabeled data has lead to growing interest in \emph{active learning}, a paradigm in which data-hungry learning algorithms adaptively select informative examples in order to lower prohibitively expensive labeling costs. Unfortunately, in standard worst-case models of learning, the active setting often provides no improvement over non-adaptive algorithms. To combat this, a series of recent works have considered a model in which the learner may ask \emph{enriched} queries beyond labels. While such models have seen success in drastically lowering label costs, they tend to come at the expense of requiring large amounts of memory. In this work, we study what families of classifiers can be learned in \emph{bounded memory}. To this end, we introduce a novel streaming-variant of enriched-query active learning along with a natural combinatorial parameter called \emph{lossless sample compression} that is sufficient for learning not only with bounded memory, but in a query-optimal and computationally efficient manner as well. Finally, we give three fundamental examples of classifier families with small, easy to compute lossless compression schemes when given access to basic enriched queries: axis-aligned rectangles, decision trees, and halfspaces in two dimensions.

Recent research has recognized interpretability and robustness as essential properties of trustworthy classification. Curiously, a connection between robustness and interpretability was empirically observed, but the theoretical reasoning behind it remained elusive. In this paper, we rigorously investigate this connection. Specifically, we focus on interpretation using decision trees and robustness to l_{\infty}-perturbation. Previous works defined the notion of r-separation as a sufficient condition for robustness. We prove upper and lower bounds on the tree size in case the data is r-separated. We then show that a tighter bound on the size is possible when the data is linearly separated. We provide the first algorithm with provable guarantees both on robustness, interpretability, and accuracy in the context of decision trees. Experiments confirm that our algorithm yields classifiers that are both interpretable and robust and have high accuracy.

We study k-median clustering under the sequential no-substitution setting. In this setting, a data stream is sequentially observed, and some of the points are selected by the algorithm as cluster centers. However, a point can be selected as a center only immediately after it is observed, before observing the next point. In addition, a selected center cannot be substituted later. We give the first algorithm for this setting that obtains a constant approximation factor on the optimal cost under a random arrival order, an exponential improvement over previous work. This is also the first constant approximation guarantee that holds without any structural assumptions on the input data. Moreover, the number of selected centers is only quasi-linear in k. Our algorithm and analysis are based on a careful cost estimation that avoids outliers, a new concept of a linear bin division, and a multi-scale approach to center selection.

Many clustering algorithms lead to cluster assignments that are hard to explain, partially because they depend on all the features of the data in a complicated way. To improve interpretability, we consider using a small decision tree to partition a data set into clusters, so that clusters can be characterized in a straightforward manner. We study this problem from a theoretical viewpoint, measuring cluster quality by the k-means and k-medians objectives. In terms of negative results, we show that popular top-down decision tree algorithms may lead to clusterings with arbitrarily large cost, and any clustering based on a tree with k leaves must incur an Omega(log k) approximation factor compared to the optimal clustering. On the positive side, for two means/medians, we show that a single threshold cut can achieve a constant factor approximation, and we give nearly-matching lower bounds; for general k > 2, we design an efficient algorithm that leads to an O(k) approximation to the optimal k-medians and an O(k^2) approximation to the optimal k-means. Prior to our work, no algorithms were known with provable guarantees independent of dimension and input size.

Combinatorial dimensions play an important role in the theory of machine learning. For example, VC dimension characterizes PAC learning, SQ dimension characterizes weak learning with statistical queries, and Littlestone dimension characterizes online learning. In this paper we aim to develop combinatorial dimensions that characterize bounded memory learning. We propose a candidate solution for the case of realizable strong learning under a known distribution, based on the SQ dimension of neighboring distributions. We prove both upper and lower bounds for our candidate solution, that match in some regime of parameters. This is the first characterization of strong learning under space constraints in any regime. In this parameter regime there is an equivalence between bounded memory and SQ learning. We conjecture that our characterization holds in a much wider regime of parameters.

With any hypothesis class one can associate a bipartite graph whose vertices are the hypotheses H on one side and all possible labeled examples X on the other side, and an hypothesis is connected to all the labeled examples that are consistent with it. We call this graph the hypotheses graph. We prove that any hypothesis class whose hypotheses graph is mixing cannot be learned using less than Omega(log^2 |H|) memory bits unless the learner uses at least a large number |H|^Omega(1) labeled examples. Our work builds on a combinatorial framework that we suggested in a previous work for proving lower bounds on space bounded learning. The strong lower bound is obtained by defining a new notion of pseudorandomness, the entropy sampler. Raz obtained a similar result using different ideas.

One can learn any hypothesis class H with O(log |H|) labeled examples. Alas, learning with so few examples requires saving the examples in memory, and this requires |X|^(O(log|H|)) memory states, where X is the set of all labeled examples. This motivates the question of how many labeled examples are needed in case the memory is bounded. Previous work showed, using techniques such as linear algebra and Fourier analysis, that parities cannot be learned with bounded memory and less than |H|^(Omega(1)) examples. One might wonder whether a general combinatorial condition exists for unlearnability with bounded memory, as we have with the condition VCdim(H) = Infinity for PAC unlearnability. In this paper we give such a condition. We show that if an hypothesis class H, when viewed as a bipartite graph between hypotheses H and labeled examples X, is mixing, then learning it requires |H|^(Omega(1)) examples under a certain bound on the memory. Note that the class of parities is mixing. Moreover, as an immediate corollary, we get that most hypothesis classes are unlearnable with bounded memory. Our proof technique is combinatorial in nature and very different from previous analyses.

We suggest a unified view of two known prediction algorithms: Context Tree Weighting (CTW) and Prediction Suffix Tree (PST), by formulating them as information limited control problems. Using a unified view of planning and information gathering we suggest a new algorithm that combines the advantages of these two extreme algorithms and interpolates efficiently between them. The unified view is based on recent ideas from optimal control under information constraints.