We define and study the problem of predicting the solution to a linear program (LP) given only partial information about its objective and constraints. This generalizes the problem of learning to predict the purchasing behavior of a rational agent who has an unknown objective function, that has been studied under the name "Learning from Revealed Preferences. We give mistake bound learning algorithms in two settings: in the first, the objective of the LP is known to the learner but there is an arbitrary, fixed set of constraints which are unknown. Each example is defined by an additional known constraint and the goal of the learner is to predict the optimal solution of the LP given the union of the known and unknown constraints. This models the problem of predicting the behavior of a rational agent whose goals are known, but whose resources are unknown. In the second setting, the objective of the LP is unknown, and changing in a controlled way. The constraints of the LP may also change every day, but are known. An example is given by a set of constraints and partial information about the objective, and the task of the learner is again to predict the optimal solution of the partially known LP.

Theitemsbecomeavailable over a sequence of rounds, and every item must be allocated immediately and irrevocably before the next one arrives.

Recent works in learning-integrated optimization have shown promise in settings where the optimization problem is only partially observed or where general-purpose optimizers perform poorly without expert tuning. By learning an optimizer $\mathbf{g}$ to tackle these challenging problems with $f$ as the objective, the optimization process can be substantially accelerated by leveraging past experience. The optimizer can be trained with supervision from known optimal solutions or implicitly by optimizing the compound function $f\circ \mathbf{g}$. The implicit approach may not require optimal solutions as labels and is capable of handling problem uncertainty; however, it is slow to train and deploy due to frequent calls to optimizer $\mathbf{g}$ during both training and testing. The training is further challenged by sparse gradients of $\mathbf{g}$, especially for combinatorial solvers.

We consider the fundamental problem of fairly and efficiently allocating $T$ indivisible items among $n$ agents with additive preferences. The items become available over a sequence of rounds, and every item must be allocated immediately and irrevocably before the next one arrives. Previous work shows that when the agents' valuations for the items are drawn from known distributions, it is possible (under mild technical assumptions) to find allocations that are envy-free with high probability and Pareto efficient ex-post. We study a \emph{partial-information} setting, where it is possible to elicit ordinal but not cardinal information. When a new item arrives, the algorithm can query each agent for the relative rank of this item with respect to a subset of the past items. When values are drawn from i.i.d.\ distributions, we give an algorithm that is envy-free and $(1-\epsilon)$-welfare-maximizing with high probability. We provide similar guarantees (envy-freeness and a constant approximation to welfare with high probability) even with minimally expressive queries that ask for a comparison to a single previous item. For independent but non-identical agents, we obtain envy-freeness and a constant approximation to Pareto efficiency with high probability. We prove that all our results are asymptotically tight.

We define and study the problem of predicting the solution to a linear program (LP) given only partial information about its objective and constraints. This generalizes the problem of learning to predict the purchasing behavior of a rational agent who has an unknown objective function, that has been studied under the name "Learning from Revealed Preferences. We give mistake bound learning algorithms in two settings: in the first, the objective of the LP is known to the learner but there is an arbitrary, fixed set of constraints which are unknown. Each example is defined by an additional known constraint and the goal of the learner is to predict the optimal solution of the LP given the union of the known and unknown constraints. This models the problem of predicting the behavior of a rational agent whose goals are known, but whose resources are unknown. In the second setting, the objective of the LP is unknown, and changing in a controlled way. The constraints of the LP may also change every day, but are known. An example is given by a set of constraints and partial information about the objective, and the task of the learner is again to predict the optimal solution of the partially known LP.

If the paper is accepted as-is, its scope should be made clear in the abstract. Q2: Please summarize your review in 1-2 sentences The paper provides valuable insight into the task of network clustering, under the SBM model with either limited information about the network, or in the streaming setting. However, the lack of even a basic simulation study, coupled with the presence of tuning parameters in Algorithms 3 and 4, lead me to question if the presented algorithms are actually practical.

We define and study the problem of predicting the solution to a linear program (LP) given only partial information about its objective and constraints. This generalizes the problem of learning to predict the purchasing behavior of a rational agent who has an unknown objective function, that has been studied under the name "Learning from Revealed Preferences". We give mistake bound learning algorithms in two settings: in the first, the objective of the LP is known to the learner but there is an arbitrary, fixed set of constraints which are unknown. Each example is defined by an additional known constraint and the goal of the learner is to predict the optimal solution of the LP given the union of the known and unknown constraints. This models the problem of predicting the behavior of a rational agent whose goals are known, but whose resources are unknown.

In this paper, we consider sparse networks consisting of a finite number of nonoverlapping communities, i.e. disjoint clusters, so that there is higher density within clusters than across clusters. Both the intra-and inter-cluster edge densities vanish when the size of the graph grows large, making the cluster reconstruction problem nosier and hence difficult to solve. We are interested in scenarios where the network size is very large, so that the adjacency matrix of the graph is hard to manipulate and store. The data stream model in which columns of the adjacency matrix are revealed sequentially constitutes a natural framework in this setting. For this model, we develop two novel clustering algorithms that extract the clusters asymptotically accurately. The first algorithm is offline, as it needs to store and keep the assignments of nodes to clusters, and requires a memory that scales linearly with the network size. The second algorithm is online, as it may classify a node when the corresponding column is revealed and then discard this information. This algorithm requires a memory growing sub-linearly with the network size. To construct these efficient streaming memory-limited clustering algorithms, we first address the problem of clustering with partial information, where only a small proportion of the columns of the adjacency matrix is observed and develop, for this setting, a new spectral algorithm which is of independent interest.