In a linear combinatorial optimization problem, we are given a family F ⊆ 2U of feasible subsets of a ground set U of n elements, and our goal is to find S* = argminS ∈ F ⟨ w,1S ⟩. Traditionally, we are either given the weight vector up-front, or else we are given a value oracle which allows us to evaluate w(S) := ⟨ w, 1S ⟩ for any S ∈ F. We consider the weaker and more robust comparison oracle, which for any two feasible sets S, T ∈ F, reveals only if w(S) is less than/equal to/greater than w(T). We ask: When can we find the optimal feasible set S* = argminS ∈ F w(S) using a small number of comparison queries? If so, when can we do this efficiently?

We provide an algorithm with regret $O(C d \log \log T)$ for contextual pricing with $C$ corrupted rounds, improving over the previous bound of $O(d^3 C \log^2(T))$ of Krishnamurthy et al. The result is based on a reduction that calls the uncorrupted algorithm as a black-box, unlike the previous approach that modifies the inner workings of the uncorrupted algorithm. As a result, it leads to a conceptually simpler algorithm. Finally, we provide a lower bound ruling out a $O(C + d\log \log T)$ algorithm. This shows that robustifying contextual pricing is harder than robustifying contextual search with $\epsilon$-ball losses, for which it is possible to design algorithms where corruptions add only an extra additive term $C$ to the regret.

Preventing the performance decay of Transformers on inputs longer than those used for training has been an important challenge in extending the context length of these models. Though the Transformer architecture has fundamentally no limits on the input sequence lengths it can process, the choice of position encoding used during training can limit the performance of these models on longer inputs. We propose a novel functional relative position encoding with progressive interpolation, FIRE, to improve Transformer generalization to longer contexts. We theoretically prove that this can represent some of the popular relative position encodings, such as T5's RPE, Alibi, and Kerple. We next empirically show that FIRE models have better generalization to longer contexts on both zero-shot language modeling and long text benchmarks.

Real-life contractual relations typically involve repeated interactions between the principal and agent, where, despite theoretical appeal, players rarely use complex dynamic strategies and instead manage uncertainty through learning algorithms. In this paper, we initiate the study of repeated contracts with learning agents, focusing on those achieving no-regret outcomes. For the canonical setting where the agent’s actions result in success or failure, we present a simple, optimal solution for the principal: Initially provide a linear contract with scalar $\alpha > 0$, then switch to a zero-scalar contract. This shift causes the agent to “free-fall” through their action space, yielding non-zero rewards for the principal at zero cost. Interestingly, despite the apparent exploitation, there are instances where our dynamic contract can make \emph{both} players better off compared to the best static contract. We then broaden the scope of our results to general linearly-scaled contracts, and, finally, to the best of our knowledge, we provide the first analysis of optimization against learning agents with uncertainty about the time horizon.

We study the design of prior-independent auctions in a setting with heterogeneous bidders. In particular, we consider the setting of selling to n bidders whose values are drawn from n independent but not necessarily identical distributions. We work in the robust auction design regime, where we assume the seller has no knowledge of the bidders’ value distributions and must design a mechanism that is prior-independent. While there have been many strong results on prior-independent auction design in the i.i.d. setting, not much is known for the heterogeneous setting, even though the latter is of significant practical importance. Unfortunately, no prior-independent mechanism can hope to always guarantee any approximation to Myerson's revenue in the heterogeneous setting; similarly, no prior-independent mechanism can consistently do better than the second-price auction. In light of this, we design a family of (parametrized) randomized auctions which approximates at least one of these benchmarks: For heterogeneous bidders with regular value distributions, our mechanisms either achieve a good approximation of the expected revenue of an optimal mechanism (which knows the bidders’ distributions) or exceeds that of the second-price auction by a certain multiplicative factor. The factor in the latter case naturally trades off with the approximation ratio of the former case. We show that our mechanism is optimal for such a trade-off between the two cases by establishing a matching lower bound. Our result extends to selling k identical items to heterogeneous bidders with an additional O(ln2 k)-factor in our trade-off between the two cases.

We consider the problem of designing prior-free revenue-maximizing mechanisms for allocating items to n buyers when the mechanism is additionally provided with an estimate for the optimal welfare (which is guaranteed to be correct to within a multiplicative factor of 1/α). In the digital goods setting (where we can allocate items to an arbitrary subset of the buyers), we demonstrate a mechanism that achieves revenue that is O(log n/α)-competitive with the optimal welfare. In the public goods setting (where we either must allocate the item to all buyers or to no buyers), we demonstrate a mechanism which is O(n log 1/α) competitive. In both settings, we show the dependence on α and n is tight. Finally, we discuss generalizations to broader classes of allocation constraints.

Inspired by applications in pricing and contract design, we study the maximization of one-sided Lipschitz functions, which only provide the (weaker) guarantee that they do not grow too quickly in one direction. We show that it is possible to learn a maximizer for such a function while incurring $O(\log \log T)$ total regret (with a universal constant independent of the number of discontinuities / complexity of the function). This regret bound is asymptotically optimal in $T$ due to a lower bound of Kleinberg and Leighton. By applying this algorithm, we show that one can sell digital goods to multiple buyers and learn the optimal linear contract in the principal-agent setting while incurring at most $O(\log \log T)$ regret.

We study the power of menus of contracts in principal-agent problems with adverse selection (agents can be one of several types) and moral hazard (we cannot observe agent actions directly). For principal-agent problems with T types and n actions, we show that the best menu of contracts can obtain a factor Ω (max(n, log T)) more utility for the principal than the best individual contract, partially resolving an open question of Guruganesh et al. [2021]. We then turn our attention to randomized menus of linear contracts, where we likewise show that randomized linear menus can be Ω(T) better than the best single linear contract. As a corollary, we show this implies an analogous gap between deterministic menus of (general) contracts and randomized menus of contracts (as introduced by Castiglioni et al. [2022]).

In this work, we investigate the problem of how to bid in repeated contextual first price auctions for a single learner (the bidder). Concretely, at each time t, the learner receives a context and decides the bid based on historical information and xt. We assume that the maximum bid of all the others follows a linear model, i.e., mt = ⟨α0, xt⟩ + zt, where is unknown to the learner and zt is randomly sampled from a noise distribution with log-concave density. In this work, we consider both binary feedback (the learner can only observe whether she wins or not) and full information feedback (the learner can observe mt) models of the learner. For binary feedback, when the noise distribution is (partially) known, we propose a bidding algorithm that achieves at most regret, where Δ is a constant reserve in first price auctions.1 For the full information feedback with unknown noise distribution, we provide an algorithm that achieves regret at most . Our approach combines an estimator for log-concave density functions and the Maximum Likelihood Estimation (MLE) method to learn the noise distribution and linear weight α0 simultaneously. We complement our results with a lower bound result such that any bidding policy in a broad class must achieve regret at least , even when the learner receives the full information feedback and is known.

We study the mechanism design problem of selling $k$ items to unit-demand buyers with private valuations for the items. A buyer either participates directly in the auction or is represented by an intermediary, who represents a subset of buyers. Our goal is to design robust mechanisms that are independent of the demand structure (i.e. how the buyers are partitioned across intermediaries), and perform well under a wide variety of possible contracts between intermediaries and buyers. We first study the case of $k$ identical items where each buyer draws its private valuation for an item i.i.d. from a known $λ$-regular distribution. We construct a robust mechanism that, independent of the demand structure and under certain conditions on the contracts between intermediaries and buyers, obtains a constant factor of the revenue that the mechanism designer could obtain had she known the buyers' valuations. In other words, our mechanism's expected revenue achieves a constant factor of the optimal welfare, regardless of the demand structure. Our mechanism is a simple posted-price mechanism that sets a take-it-or-leave-it per-item price that depends on $k$ and the total number of buyers, but does not depend on the demand structure or the downstream contracts. Next we generalize our result to the case when the items are not identical. We assume that the item valuations are separable. For this case, we design a mechanism that obtains at least a constant fraction of the optimal welfare, by using a menu of posted prices. This mechanism is also independent of the demand structure, but makes a relatively stronger assumption on the contracts between intermediaries and buyers, namely that each intermediary prefers outcomes with a higher sum of utilities of the subset of buyers represented by it.

We revisit the problem of learning mixtures of spherical Gaussians. Given samples from a mixture $\frac{1}{k}\sum_{j=1}^{k}\mathcal{N}(\mu_j, I_d)$, the goal is to estimate the means $\mu_1, \mu_2, \ldots, \mu_k \in \mathbb{R}^d$ up to a small error. The hardness of this learning problem can be measured by the \emph{separation} $\Delta$ defined as the minimum distance between all pairs of means. Regev and Vijayaraghavan (2017) showed that with $\Delta = \Omega(\sqrt{\log k})$ separation, the means can be learned using $\mathrm{poly}(k, d)$ samples, whereas super-polynomially many samples are required if $\Delta = o(\sqrt{\log k})$ and $d = \Omega(\log k)$. This leaves open the low-dimensional regime where $d = o(\log k)$. In this work, we give an algorithm that efficiently learns the means in $d = O(\log k/\log\log k)$ dimensions under separation $d/\sqrt{\log k}$ (modulo doubly logarithmic factors). This separation is strictly smaller than $\sqrt{\log k}$, and is also shown to be necessary. Along with the results of Regev and Vijayaraghavan (2017), our work almost pins down the critical separation threshold at which efficient parameter learning becomes possible for spherical Gaussian mixtures. More generally, our algorithm runs in time $\mathrm{poly}(k)\cdot f(d, \Delta, \epsilon)$, and is thus fixed-parameter tractable in parameters $d$, $\Delta$ and $\epsilon$. Our approach is based on estimating the Fourier transform of the mixture at carefully chosen frequencies, and both the algorithm and its analysis are simple and elementary. Our positive results can be easily extended to learning mixtures of non-Gaussian distributions, under a mild condition on the Fourier spectrum of the distribution.

Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [ 31 ] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut , for many others, e.g., Max-SAT and Max-DiCut , the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut , Max-2SAT , and Max-DiCut , and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.

The connection between games and no-regret algorithms has been widely studied in the literature. A fundamental result is that when all players play no-regret strategies, this produces a sequence of actions whose time-average is a coarse-correlated equilibrium of the game. However, much less is known about equilibrium selection in the case that multiple equilibria exist. In this work, we study the convergence of no-regret bidding algorithms in auctions. Besides being of theoretical interest, bidding dynamics in auctions is an important question from a practical viewpoint as well. We study the repeated game between bidders in which a single item is sold at each time step and the bidder's value is drawn from an unknown distribution. We show that if the bidders use any mean-based learning rule then the bidders converge with high probability to the truthful pure Nash Equilibrium in a second price auction, in VCG auction in the multi-slot setting and to the Bayesian Nash equilibrium in a first price auction. We note mean-based algorithms cover a wide variety of known no-regret algorithms such as Exp3, UCB, \epsilon-Greedy etc. Also, we analyze the convergence of the individual iterates produced by such learning algorithms, as opposed to the time-average of the sequence. Our experiments corroborate our theoretical findings and also find a similar convergence when we use other strategies such as Deep Q-Learning.

The applicability of agglomerative clustering, for inferring both hierarchical and flat clustering, is limited by its scalability. Existing scalable hierarchical clustering methods sacrifice quality for speed and often lead to over-merging of clusters. In this paper, we present a scalable, agglomerative method for hierarchical clustering that does not sacrifice quality and scales to billions of data points. We perform a detailed theoretical analysis, showing that under mild separability conditions our algorithm can not only recover the optimal flat partition but also provide a two-approximation to non-parametric DP-Means objective. This introduces a novel application of hierarchical clustering as an approximation algorithm for the non-parametric clustering objective. We additionally relate our algorithm to the classic hierarchical agglomerative clustering method. We perform extensive empirical experiments in both hierarchical and flat clustering settings and show that our proposed approach achieves state-of-the-art results on publicly available clustering benchmarks. Finally, we demonstrate our method's scalability by applying it to a dataset of 30 billion queries. Human evaluation of the discovered clusters show that our method finds better quality of clusters than the current state-of-the-art.

We consider the following variant of contextual linear bandits motivated by routing applications in navigational engines and recommendation systems. We wish to learn a hidden $d$-dimensional value $w^*$. Every round, we are presented with a subset $\mathcal{X}_t \subseteq \mathbb{R}^d$ of possible actions. If we choose (i.e. recommend to the user) action $x_t$, we obtain utility $\langle x_t, w^* \rangle$ but only learn the identity of the best action $\arg\max_{x \in \X_t} \langle x, w^* \rangle$.We design algorithms for this problem which achieve regret $O(d\log T)$ and $\exp(O(d \log d))$. To accomplish this, we design novel cutting-plane algorithms with low “regret” -- the total distance between the true point $w^*$ and the hyperplanes the separation oracle returns. We also consider the variant where we are allowed to provide a list of several recommendations. In this variant, we give an algorithm with $O(d^2 \log d)$ regret and list size $\poly(d)$. Finally, we construct nearly tight algorithms for a weaker variant of this problem where the learner only learns the identity of an action that is better than the recommendation. Our results rely on new algorithmic techniques in convex geometry (including a variant of Steiner’s formula for the centroid of a convex set) which may be of independent interest.

We consider the problem of multi-class classification, where a stream of adversarially chosen queries arrive and must be assigned a label online. Unlike traditional bounds which seek to minimize the misclassification rate, we minimize the total distance from each query to the region corresponding to its assigned label. When the true labels are determined via a nearest neighbor partition -- i.e. the label of a point is given by which of $k$ centers it is closest to in Euclidean distance -- we show that one can achieve a loss that is independent of the total number of queries. We complement this result by showing that learning general convex sets requires an almost linear loss per query. Our results build off of regret guarantees for the problem of contextual search. In addition, we develop a novel reduction technique from multiclass classification to binary classification which may be of independent interest.

In the classical principal-agent problem, a principal must design a contract to incentivize an agent to perform an action on behalf of the principal. We study the classical principal-agent problem in a setting where the agent can be of one of several types (affecting the outcome of actions they might take). This combines the contract theory phenomena of "moral hazard" (incomplete information about actions) with that of "adverse selection" (incomplete information about types).

The problem of chasing convex functions is easy to state: faced with a sequence of convex functions f t over d-dimensional Euclidean spaces, the goal of the algorithm is to output a point x t at each time, so that the sum of the function costs f t (x t ), plus the movement costs ||x t − x t − 1 || is minimized. This problem generalizes questions in online algorithms such as caching and the k-server problem. In 1994, Friedman and Linial posed the question of getting an algorithm with a competitive ratio that depends only on the dimension d. In this talk we give an O (d)-competitive algorithm, based on the notion of the Steiner point of a convex body.

We study the problem of chasing convex bodies online: given a sequence of convex bodies the algorithm must respond with points in an online fashion (i.e., is chosen before is revealed). The objective is to minimize the sum of distances between successive points in this sequence. Bubeck et al. (STOC 2019) gave a -competitive algorithm for this problem. We give an algorithm that is -competitive for any sequence of length .

We consider the problem of chasing convex functions, where functions arrive over time. The player takes actions after seeing the function, and the goal is to achieve a small function cost for these actions, as well as a small cost for moving between actions. While the general problem requires a polynomial dependence on the dimension, we show how to get dimension-independent bounds for well-behaved functions. In particular, we consider the case where the convex functions are $\kappa$-well-conditioned, and give an algorithm that achieves an $O(\sqrt \kappa)$-competitiveness. Moreover, when the functions are supported on $k$-dimensional affine subspaces—e.g., when the function are the indicators of some affine subspaces—we get $O(\min(k, \sqrt{k \log T}))$-competitive algorithms for request sequences of length $T$. We also show some lower bounds, that well-conditioned functions require $\Omega(\kappa^{1/3})$-competitiveness, and $k$-dimensional functions require $\Omega(\sqrt{k})$-competitiveness.

Transformers-based models, such as BERT, have been one of the most successful deep learning models for NLP. Unfortunately, one of their core limitations is the quadratic dependency (mainly in terms of memory) on the sequence length due to their full attention mechanism. To remedy this, we propose, BigBird, a sparse attention mechanism that reduces this quadratic dependency to linear. We show that BigBird is a universal approximator of sequence functions and is Turing complete, thereby preserving these properties of the quadratic, full attention model. Along the way, our theoretical analysis reveals some of the benefits of having $O(1)$ global tokens (such as CLS), that attend to the entire sequence as part of the sparse attention mechanism. The proposed sparse attention can handle sequences of length up to 8x of what was previously possible using similar hardware. As a consequence of the capability to handle longer context, BigBird drastically improves performance on various NLP tasks such as question answering and summarization. We also propose novel applications to genomics data.

We study the problem of chasing convex bodies online: given a sequence of convex bodies Kt ⊆ ℝd the algorithm must respond with points xt ϵ Kt in an on-line fashion (i.e., xt is chosen before Kt+1 is revealed). The objective is to minimize the total distance between successive points in this sequence. Recently, Bubeck et al. (STOC 2019) gave a 2O(d)-competitive algorithm for this problem. We give an algorithm that is -competitive for any sequence of length T.

Semi-definite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [23] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.

We study the minimum-cost metric perfect matching problem under online i.i.d arrivals. We are given a fixed metric with a server at each of the points, and then requests arrive online, each drawn independently from a known probability distribution over the points. Each request has to be matched to a free server, with cost equal to the distance. The goal is to minimize the expected total cost of the matching. Such stochastic arrival models have been widely studied for the maximization variants of the online matching problem; however, the only known result for the minimization problem is a tight $O(\log n)$-competitiveness for the random-order arrival model. This is in contrast with the adversarial model, where an optimal competitive ratio of $O(\log n)$ has long been conjectured and remains a tantalizing open question. In this paper, we show improved results in the i.i.d arrival model. We show how the i.i.d model can be used to give substantially better algorithms: our main result is an $O((\log \log \log n)^2)$-competitive algorithm in this model. Along the way we give a $9$-competitive algorithm for the line and tree metrics. Both results imply a strict separation between the i.i.d model and the adversarial and random order models, both for general metrics and these much-studied metrics.

We study the classic bin packing problem in a fully-dynamic setting, where new items can arrive and old items may depart. We want algorithms with low asymptotic competitive ratio while repacking items sparingly between updates. Formally, each item i has a movement cost c_i >= 0, and we want to use alpha * OPT bins and incur a movement cost gamma * c_i, either in the worst case, or in an amortized sense, for alpha, gamma as small as possible. We call gamma the recourse of the algorithm. This is motivated by cloud storage applications, where fully-dynamic bin packing models the problem of data backup to minimize the number of disks used, as well as communication incurred in moving file backups between disks. Since the set of files changes over time, we could recompute a solution periodically from scratch, but this would give a high number of disk rewrites, incurring a high energy cost and possible wear and tear of the disks. In this work, we present optimal tradeoffs between number of bins used and number of items repacked, as well as natural extensions of the latter measure.

We consider the maximum independent set problem on sparse graphs with maximum degree $d$. We show that the Lovász $\vartheta$-function based semidefinite program (SDP) has an integrality gap of $ \widetilde{O}(d/\log^{3/2} d)$, improving on the previous best result of $\widetilde{O}(d/\log d)$. This improvement is based on a new Ramsey-theoretic bound on the independence number of $K_r$-free graphs for large values of $r$. We also show that for stronger SDPs, namely, those obtained using poly$\log(d)$ levels of the ${SA}^{+}$ semidefinite hierarchy, the integrality gap reduces to $\widetilde{O}(d/\log^2 d)$. This matches the best unique-games-based hardness result up to lower-order poly$(\log\log d)$ factors. Finally, we give an algorithmic version of this ${SA}^{+}$-based integrality gap result, albeit using $d$ levels of ${SA}^{+}$, via a coloring algorithm of Johansson.

We study a generalization of the Steiner tree problem, where we are given a weighted network G together with a collection of k subsets of its vertices and a root r. We wish to construct a minimum cost network such that the network supports one unit of flow to the root from every node in a subset simultaneously. The network constructed does not need to support flows from all the subsets simultaneously. We settle an open question regarding the complexity of this problem for k=2, and give a 3/2-approximation algorithm that improves over a (trivial) known 2-approximation. Furthermore, we prove some structural results that prevent many well-known techniques from doing better than the known O(log n)-approximation. Despite these obstacles, we conjecture that this problem should have an O(1)-approximation. We also give an approximation result for a variant of the problem where the solution is required to be a path.

In the aversion k-clustering problem, given a metric space, we want to cluster the points into k clusters. The cost incurred by each point is the distance to the furthest point in its cluster, and the cost of the clustering is the sum of all these per-point-costs. This problem is motivated by questions in generating automatic abstractions of extensive-form games. We reduce this problem to a "local" k-median problem where each facility has a prescribed radius and can only connect to clients within that radius. Our main results is a constant-factor approximation algorithm for the aversion k-clustering problem via the local k-median problem. We use a primal-dual approach; our technical contribution is a non-local rounding step which we feel is of broader interest.

We study the k-route generalizations of various cut problems, the most general of which is k-route multicut (k-MC) problem, wherein we have r source-sink pairs and the goal is to delete a minimum-cost set of edges to reduce the edge-connectivity of every source-sink pair to below k. The k-route extensions of multiway cut (k-MWC), and the minimum s-t cut problem (k-(s, t)-Cut), are similarly defined. We present various approximation and hardness results for k-MC, k-MWC, and k-(s,t)-Cut that improve the state-of-the-art for these problems in several cases. Our contributions are threefold. For k-route multiway cut, we devise simple, but surprisingly effective, combinatorial algorithms that yield bicriteria approximation guarantees that markedly improve upon the previous-best guarantees. For k-route multicut, we design algorithms that improve upon the previous-best approximation factors by roughly an -factor, when k = 2, and for general k and unit costs and any fixed violation of the connectivity threshold k. The main technical innovation is the definition of a new, powerful region growing lemma that allows us to perform region-growing in a recursive fashion even though the LP solution yields a different metric for each source-sink pair, and without incurring an O(log2 r) blow-up in the cost that is inherent in some previous applications of region growing to k-route cuts. We obtain the same benefits as [15] do in their divide-and-conquer algorithms, and thereby obtain an O(ln r ln ln r)-approximation to the cost. We also obtain some extensions to k-route node-multicut problems. We complement these results by showing that the k-route s-t cut problem is at least as hard to approximate as the densest-k-subgraph (DkS) problem on uniform hypergraphs. In particular, this implies that one cannot avoid a poly(k)-factor if one seeks a unicriterion approximation, without improving the state-of-the-art for DkS on graphs, and proving the existence of a family of one-way functions. Previously, only NP-hardness of k-(s; t)-Cut was known.

We consider the maximum independent set problem on graphs with maximum degree d. We show that the integrality gap of the Lovasz Theta function-based SDP has an integrality gap of O~(d/log3/2 d). This improves on the previous best result of O~(d/log d), and narrows the gap of this basic SDP to the integrality gap of O~(d/log2 d) recently shown for stronger SDPs, namely those obtained using poly log(d) levels of the SA+ semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of Kr-free graphs for large values of r.