In this work, we study longest common substring, pattern matching, and wildcard pattern matching in the asymmetric streaming model. In this streaming model, we have random access to one string and streaming access to the other one. We present streami
ng algorithms with provable guarantees for these three fundamental problems. In particular, our algorithms for pattern matching improve the upper bound and beat the unconditional lower bounds on the memory of randomized and deterministic streaming algorithms. In addition to this, we present algorithms for wildcard pattern matching in the asymmetric streaming model that have optimal space and time.
Understanding generalization and estimation error of estimators for simple models such as linear and generalized linear models has attracted a lot of attention recently. This is in part due to an interesting observation made in machine learning commu
nity that highly over-parameterized neural networks achieve zero training error, and yet they are able to generalize well over the test samples. This phenomenon is captured by the so called double descent curve, where the generalization error starts decreasing again after the interpolation threshold. A series of recent works tried to explain such phenomenon for simple models. In this work, we analyze the asymptotics of estimation error in ridge estimators for convolutional linear models. These convolutional inverse problems, also known as deconvolution, naturally arise in different fields such as seismology, imaging, and acoustics among others. Our results hold for a large class of input distributions that include i.i.d. features as a special case. We derive exact formulae for estimation error of ridge estimators that hold in a certain high-dimensional regime. We show the double descent phenomenon in our experiments for convolutional models and show that our theoretical results match the experiments.
We study the problem of machine unlearning and identify a notion of algorithmic stability, Total Variation (TV) stability, which we argue, is suitable for the goal of exact unlearning. For convex risk minimization problems, we design TV-stable algori
thms based on noisy Stochastic Gradient Descent (SGD). Our key contribution is the design of corresponding efficient unlearning algorithms, which are based on constructing a (maximal) coupling of Markov chains for the noisy SGD procedure. To understand the trade-offs between accuracy and unlearning efficiency, we give upper and lower bounds on excess empirical and populations risk of TV stable algorithms for convex risk minimization. Our techniques generalize to arbitrary non-convex functions, and our algorithms are differentially private as well.
We study the online discrepancy minimization problem for vectors in $mathbb{R}^d$ in the oblivious setting where an adversary is allowed fix the vectors $x_1, x_2, ldots, x_n$ in arbitrary order ahead of time. We give an algorithm that maintains $O(s
qrt{log(nd/delta)})$ discrepancy with probability $1-delta$, matching the lower bound given in [Bansal et al. 2020] up to an $O(sqrt{log log n})$ factor in the high-probability regime. We also provide results for the weighted and multi-col
We address the following dynamic version of the school choice question: a city, named City, admits students in two temporally-separated rounds, denoted $mathcal{R}_1$ and $mathcal{R}_2$. In round $mathcal{R}_1$, the capacity of each school is fixed a
nd mechanism $mathcal{M}_1$ finds a student optimal stable matching. In round $mathcal{R}_2$, certain parameters change, e.g., new students move into the City or the City is happy to allocate extra seats to specific schools. We study a number of Settings of this kind and give polynomial time algorithms for obtaining a stable matching for the new situations. It is well established that switching the school of a student midway, unsynchronized with her classmates, can cause traumatic effects. This fact guides us to two types of results, the first simply disallows any re-allocations in round $mathcal{R}_2$, and the second asks for a stable matching that minimizes the number of re-allocations. For the latter, we prove that the stable matchings which minimize the number of re-allocations form a sublattice of the lattice of stable matchings. Observations about incentive compatibility are woven into these results. We also give a third type of results, namely proofs of NP-hardness for a mechanism for round $mathcal{R}_2$ under certain settings.
We are given a stable matching instance $A$ and a set $S$ of errors that can be introduced into $A$. Each error consists of applying a specific permutation to the preference list of a chosen boy or a chosen girl. Assume that $A$ is being transmitted
over a channel which introduces one error from set $S$; as a result, the channel outputs this new instance. We wish to find a matching that is stable for $A$ and for each of the $|S|$ possible new instances. If $S$ is picked from a special class of errors, we give an $O(|S| p(n))$ time algorithm for this problem. We call the obtained matching a fully robust stable matching w.r.t. $S$. In particular, if $S$ is polynomial sized, then our algorithm runs in polynomial time. Our algorithm is based on new, non-trivial structural properties of the lattice of stable matchings; these properties pertain to certain join semi-sublattices of the lattice. Birkhoffs Representation Theorem for finite distributive lattices plays a special role in our algorithms.
We study the problem of finding solutions to the stable matching problem that are robust to errors in the input and we obtain a polynomial time algorithm for a special class of errors. In the process, we also initiate work on a new structural questio
n concerning the stable matching problem, namely finding relationships between the lattices of solutions of two nearby instances. Our main algorithmic result is the following: We identify a polynomially large class of errors, $D$, that can be introduced in a stable matching instance. Given an instance $A$ of stable matching, let $B$ be the random variable that represents the instance that results after introducing {em one} error from $D$, chosen via a given discrete probability distribution. The problem is to find a stable matching for $A$ that maximizes the probability of being stable for $B$ as well. Via new structural properties of the type described in the question stated above, we give a combinatorial polynomial time algorithm for this problem. We also show that the set of robust stable matchings for instance $A$, under probability distribution $p$, forms a sublattice of the lattice of stable matchings for $A$. We give an efficient algorithm for finding a succinct representation for this set; this representation has the property that any member of the set can be efficiently retrieved from it.
We study a natural generalization of stable matching to the maximum weight stable matching problem and we obtain a combinatorial polynomial time algorithm for it by reducing it to the problem of finding a maximum weight ideal cut in a DAG. We give th
e first polynomial time algorithm for the latter problem; this algorithm is also combinatorial. The combinatorial nature of our algorithms not only means that they are efficient but also that they enable us to obtain additional structural and algorithmic results: - We show that the set, $cal M$, of maximum weight stable matchings forms a sublattice $cal L$ of the lattice $cal L$ of all stable matchings. - We give an efficient algorithm for finding boy-optimal and girl-optimal matchings in $cal M$. - We generalize the notion of rotation, a central structural notion in the context of the stable matching problem, to meta-rotation. Just as rotations help traverse the lattice of all stable matchings, macro-rotations help traverse the sublattice over $cal M$.
Recently Cole and Gkatzelis gave the first constant factor approximation algorithm for the problem of allocating indivisible items to agents, under additive valuations, so as to maximize the Nash Social Welfare. We give constant factor algorithms for
a substantial generalization of their problem -- to the case of separable, piecewise-linear concave utility functions. We give two such algorithms, the first using market equilibria and the second using the theory of stable polynomials. In AGT, there is a paucity of methods for the design of mechanisms for the allocation of indivisible goods and the result of Cole and Gkatzelis seemed to be taking a major step towards filling this gap. Our result can be seen as another step in this direction.
With the rapid growth of the cloud computing marketplace, the issue of pricing resources in the cloud has been the subject of much study in recent years. In this paper, we identify and study a new issue: how to price resources in the cloud so that th
e customers risk is minimized, while at the same time ensuring that the provider accrues his fair share. We do this by correlating the revenue stream of the customer to the prices charged by the provider. We show that our mechanism is incentive compatible in that it is in the best interest of the customer to provide his true revenue as a function of the resources rented. We next add another restriction to the price function, i.e., that it be linear. This removes the distortion that creeps in when the customer has to pay more money for less resources. Our algorithms for both the schemes mentioned above are efficient.
