Siddharth Barman

Decomposition Methods for Large Scale LP Decoding

When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper, we draw on decomposition methods from optimization theory, specifically the alternating direction method of multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a “two-slice” characterization of the parity polytope, the polytope formed by taking the convex hull of all codewords of the single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by the ADMM decoder and its solution allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher SNR than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using the ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.

Decomposition methods for large scale LP decoding

Feldman et al. (IEEE Trans. Inform. Theory, Mar. 2005) showed that linear programming (LP) can be used to decode linear error correcting codes. The bit-error-rate performance of LP decoding is comparable to state-of-the-art BP decoders, but has significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper we draw on decomposition methods from optimization theory to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a nearly linear time algorithm for two-norm projection onto the parity polytope. This allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently.

Preventing equivalence attacks in updated, anonymized data

In comparison to the extensive body of existing work considering publish-once, static anonymization, dynamic anonymization is less well studied. Previous work, most notably m-invariance, has made considerable progress in devising a scheme that attempts to prevent individual records from being associated with too few sensitive values. We show, however, that in the presence of updates, even an m-invariant table can be exploited by a new type of attack we call the “equivalence-attack.” To deal with the equivalence attack, we propose a graph-based anonymization algorithm that leverages solutions to the classic “min-cut/max-flow” problem, and demonstrate with experiments that our algorithm is efficient and effective in preventing equivalence attacks.

A tale of two metrics: simultaneous bounds on competitiveness and regret

We consider algorithms for “smoothed online convex optimization” (SOCO) problems, which are a hybrid between online convex optimization (OCO) and metrical task system (MTS) problems. Historically, the performance metric for OCO was regret and that for MTS was competitive ratio (CR). There are algorithms with either sublinear regret or constant CR, but no known algorithm achieves both simultaneously. We show that this is a fundamental limitation – no algorithm (deterministic or randomized) can achieve sublinear regret and a constant CR, even when the objective functions are linear and the decision space is one dimensional. However, we present an algorithm that, for the important one dimensional case, provides sublinear regret and a CR that grows arbitrarily slowly.

Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Caratheodory's Theorem

We present algorithmic applications of an approximate version of Caratheodorys theorem. The theorem states that given a set of vectors X in Rd, for every vector in the convex hull of X there exists an ε-close (under the p-norm distance, for 2 ≤ p < ∞) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on ε and the norm p and is independent of the dimension d. This theorem can be derived by instantiating Maureys lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result. Using this theorem we establish that in a bimatrix game with n x n payoff matrices A, B, if the number of non-zero entries in any column of A+B is at most s then an ε-Nash equilibrium of the game can be computed in time nO(log s/ε2}). This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity s. Moreover, for arbitrary bimatrix games---since s can be at most n---the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003). The approximate Carathéodorys theorem also leads to an additive approximation algorithm for the densest k-bipartite subgraph problem. Given a graph with n vertices and maximum degree d, the developed algorithm determines a k x k bipartite subgraph with density within ε (in the additive sense) of the optimal density in time nO(log d/ε2).

On the complexity of privacy-preserving complex event processing

Complex Event Processing (CEP) Systems are stream processing systems that monitor incoming event streams in search of userspecified event patterns. While CEP systems have been adopted in a variety of applications, the privacy implications of event pattern reporting mechanisms have yet to be studied - a stark contrast to the significant amount of attention that has been devoted to privacy for relational systems. In this paper we present a privacy problem that arises when the system must support desired patterns (those that should be reported if detected) and private patterns (those that should not be revealed). We formalize this problem, which we term privacy-preserving, utility maximizing CEP (PP-CEP), and analyze its complexity under various assumptions. Our results show that this is a rich problem to study and shed some light on the difficulty of developing algorithms that preserve utility without compromising privacy.

Simple approximate equilibria in large games

We prove that in every normal form n-player game with m actions for each player, there exists an approximate Nash equilibrium in which each player randomizes uniformly among a set of O(log m + log n) pure actions. This result induces an O(N log log N)-time algorithm for computing an approximate Nash equilibrium in games where the number of actions is polynomial in the number of players (m=poly(n)); here N=nmn is the size of the game (the input size). Furthermore, when the number of actions is a fixed constant (m=O(1)) the same algorithm runs in O(Nlog log log N) time. In addition, we establish an inverse connection between the entropy of Nash equilibria in the game, and the time it takes to find such an approximate Nash equilibrium using the random sampling method. We also consider other relevant notions of equilibria. Specifically, we prove the existence of approximate correlated equilibrium of support size polylogarithmic in the number of players, n, and the number of actions per player, m. In particular, using the probabilistic method, we show that there exists a multiset of action profiles of polylogarithmic size such that the uniform distribution over this multiset forms an approximate correlated equilibrium. Along similar lines, we establish the existence of approximate coarse correlated equilibrium with logarithmic support. We complement these results by considering the computational complexity of determining small-support approximate equilibria. We show that random sampling can be used to efficiently determine an approximate coarse correlated equilibrium with logarithmic support. But, such a tight result does not hold for correlated equilibrium, i.e., sampling might generate an approximate correlated equilibrium of support size Ω(m) where m is the number of actions per player. Finally, we show that finding an exact correlated equilibrium with smallest possible support is NP-hard under Cook reductions, even in the case of two-player zero-sum games.

Approximation Algorithms for Maximin Fair Division

We consider the problem of dividing indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, that is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focussed on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 1/10-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the paper is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions.

Online Convex Optimization Using Predictions

Making use of predictions is a crucial, but under-explored, area of online algorithms. This paper studies a class of online optimization problems where we have external noisy predictions available. We propose a stochastic prediction error model that generalizes prior models in the learning and stochastic control communities, incorporates correlation among prediction errors, and captures the fact that predictions improve as time passes. We prove that achieving sublinear regret and constant competitive ratio for online algorithms requires the use of an unbounded prediction window in adversarial settings, but that under more realistic stochastic prediction error models it is possible to use Averaging Fixed Horizon Control (AFHC) to simultaneously achieve sublinear regret and constant competitive ratio in expectation using only a constant-sized prediction window. Furthermore, we show that the performance of AFHC is tightly concentrated around its mean.

Finding Any Nontrivial Coarse Correlated Equilibrium Is Hard

One of the most appealing aspects of the (coarse) correlated equilibrium concept is that natural dynamics quickly arrive at approximations of such equilibria, even in games with many players. In addition, there exist polynomial-time algorithms that compute exact (coarse) correlated equilibria. In light of these results, a natural question is how good are the (coarse) correlated equilibria that can arise from any efficient algorithm or dynamics. In this paper we address this question, and establish strong negative results. In particular, we show that in multiplayer games that have a succinct representation, it is NP-hard to compute any coarse correlated equilibrium (or approximate coarse correlated equilibrium) with welfare strictly better than the worst possible. The focus on succinct games ensures that the underlying complexity question is interesting; many multiplayer games of interest are in fact succinct. Our results imply that, while one can efficiently compute a coarse correlated equilibrium, one cannot provide any nontrivial welfare guarantee for the resulting equilibrium, unless P=NP. We show that analogous hardness results hold for correlated equilibria, and persist under the egalitarian objective or Pareto optimality. To complement the hardness results, we develop an algorithmic framework that identifies settings in which we can efficiently compute an approximate correlated equilibrium with near-optimal welfare. We use this framework to develop an efficient algorithm for computing an approximate correlated equilibrium with near-optimal welfare in aggregative games.