In pretraining data detection, the goal is to detect whether a given sentence is in the dataset used for training a Large Language Model LLM). Recent methods (such as Min-K % and Min-K%++) reveal that most training corpora are likely contaminated with both sensitive content and evaluation benchmarks, leading to inflated test set performance. These methods sometimes fail to detect samples from the pretraining data, primarily because they depend on statistics composed of causal token likelihoods. We introduce Infilling Score, a new test-statistic based on non-causal token likelihoods. Infilling Score can be computed for autoregressive models without re-training using Bayes rule. A naive application of Bayes rule scales linearly with the vocabulary size. However, we propose a ratio test-statistic whose computation is invariant to vocabulary size. Empirically, our method achieves a significant accuracy gain over state-of-the-art methods including Min-K%, and Min-K%++ on the WikiMIA benchmark across seven models with different parameter sizes. Further, we achieve higher AUC compared to reference-free methods on the challenging MIMIR benchmark. Finally, we create a benchmark dataset consisting of recent data sources published after the release of Llama-3; this benchmark provides a statistical baseline to indicate potential corpora used for Llama-3 training.

Generative models transform random noise into images, while their inversion aims to reconstruct structured noise for recovery and editing. This paper addresses two key tasks: (i) *inversion* and (ii) *editing* of real images using stochastic equivalents of rectified flow models (e.g., Flux). While Diffusion Models (DMs) dominate the field of generative modeling for images, their inversion suffers from faithfulness and editability challenges due to nonlinear drift and diffusion. Existing DM inversion methods require costly training of additional parameters or test-time optimization of latent variables. Rectified Flows (RFs) offer a promising alternative to DMs, yet their inversion remains underexplored. We propose RF inversion using dynamic optimal control derived via a linear quadratic regulator, and prove that the resulting vector field is equivalent to a rectified stochastic differential equation. We further extend our framework to design a stochastic sampler for Flux. Our method achieves state-of-the-art performance in zero-shot inversion and editing, surpassing prior works in stroke-to-image synthesis and semantic image editing, with large-scale human evaluations confirming user preference. See our project page https://rf-inversion.github.io/ for code and demo.

We propose Reference-Based Modulation (RB-Modulation), a new plug-and-play solution for training-free personalization of diffusion models. Existing training-free approaches exhibit difficulties in (a) style extraction from reference images in the absence of additional style or content text descriptions, (b) unwanted content leakage from reference style images, and (c) effective composition of style and content. RB-Modulation is built on a novel stochastic optimal controller where a style descriptor encodes the desired attributes through a terminal cost. The resulting drift not only overcomes the difficulties above, but also ensures high fidelity to the reference style and adheres to the given text prompt. We also introduce a cross-attention-based feature aggregation scheme that allows RB-Modulation to decouple content and style from the reference image. With theoretical justification and empirical evidence, our test-time optimization framework demonstrates precise extraction and control of *content* and *style* in a training-free manner. Further, our method allows a seamless composition of content and style, which marks a departure from the dependency on external adapters or ControlNets. See project page: https://rb-modulation.github.io/ for code and further details.

Unconscious bias has been shown to influence how we assess our peers, with consequences for hiring, promotions and admissions. In this work, we focus on affinity bias, the component of unconscious bias which leads us to prefer people who are similar to us, despite no deliberate intention of favoritism. In a world where the people hired today become part of the hiring committee of tomorrow, we are particularly interested in understanding (and mitigating) how affinity bias affects this feedback loop. This problem has two distinctive features: 1) we only observe the _biased value_ of a candidate, but we want to optimize with respect to their _real value_ 2) the bias towards a candidate with a specific set of traits depends on the _fraction_ of people in the hiring committee with the same set of traits. We introduce a new bandits variant that exhibits those two features, which we call affinity bandits. Unsurprisingly, classical algorithms such as UCB often fail to identify the best arm in this setting. We prove a new instance-dependent regret lower bound, which is larger than that in the standard bandit setting by a multiplicative function of $K$. Since we treat rewards that are _time-varying_ and _dependent on the policy's past actions_, deriving this lower bound requires developing proof techniques beyond the standard bandit techniques. Finally, we design an elimination-style algorithm which nearly matches this regret, despite never observing the real rewards.

Diffusion Language Models (DLMs) promise parallel generation and bidirectional context, yet they underperform autoregressive (AR) models in both *likelihood modeling* and *generated text quality*. We identify that this performance gap arises when important tokens (e.g., key words or low-frequency words that anchor a sentence) are masked early in the forward process, limiting contextual information for accurate reconstruction. To address this, we introduce the *Anchored Diffusion Language Model (ADLM)*, a novel two-stage framework that first predicts distributions over important tokens via an anchor network, and then predicts the likelihoods of missing tokens conditioned on the anchored predictions. ADLM significantly improves test perplexity on LM1B and OpenWebText, achieving up to 25.4\% gains over prior DLMs, and narrows the gap with strong AR baselines. It also achieves state-of-the-art zero-shot generalization across seven benchmarks and surpasses AR models in MAUVE score, which marks the first time a DLM generates better human-like text than an AR model. Theoretically, we derive an Anchored Negative Evidence Lower Bound (ANELBO) objective and show that anchoring improves sample complexity and likelihood modeling. Beyond diffusion, anchoring boosts performance in AR models and enhances reasoning in math and logic tasks, outperforming existing chain-of-thought approaches. Please see our project page: [anchored-diffusion-llm.github.io](https://anchored-diffusion-llm.github.io/) for code and demo.

Pandora’s Box is a fundamental stochastic optimization problem, where the decision-maker must find a good alternative, while minimizing the search cost of exploring the value of each alternative. In the original formulation, it is assumed that accurate distributions are given for the values of all the alternatives, while recent work studies the online variant of Pandora’s Box where the distributions are originally unknown. In this work, we study Pandora’s Box in the online setting, while incorporating context. At each round, we are presented with a number of alternatives each having a context, an exploration cost and an unknown value drawn from an unknown distribution that may change at every round. Our main result is a no-regret algorithm that performs comparably well against the optimal algorithm which knows all prior distributions exactly. Our algorithm works even in the bandit setting where the algorithm never learns the values of the alternatives that were not explored. The key technique that enables our result is a novel modification of the realizability condition in contextual bandits that connects a context to a sufficient statistic of each alternative’s distribution (its reservation value) rather than its mean.

Sampling from the posterior distribution in latent diffusion models for inverse problems is computationally challenging. Existing methods often rely on Tweedie's first-order moments that tend to induce biased results [32]. Second-order approximations are computationally prohibitive, making standard reverse diffusion processes in-tractable for posterior sampling. We present Second-order Tweedie sampler from Surrogate Loss (STSL), a novel sampler offering efficiency comparable to first-order Tweedie while enabling tractable reverse processes using second-order approximation. Theoretical results reveal that our approach establishes a lower bound through a surrogate loss and enables a tractable reverse process using the trace of the Hessian with only$\mathcal{O}(1)$compute. We show STSL out-performs SoTA solvers PSLD [43] and P2L [10] by reducing neural function evaluations by 4X and 8X, respectively, while enhancing sampling quality on FFHQ, ImageNet, and COCO benchmarks. Moreover, STSL extends to text-guided image editing, effectively mitigating residual distortions in corrupted images. To our best knowledge, this is the first work to offer an efficient second-order approximation for solving inverse problems using latent diffusion, which further enables editing real-world images with corruptions.

In many interactive decision-making problems, there is contextual side information that remains fixed within the course of an interaction. This problem has been studied quite extensively under the assumption the context is fully observed, as well as in the opposing limit when the context is unobserved, a special type of POMDP also referred to as a Latent MDP (LMDP). In this work, we consider a class of decision problems that interpolates between the settings, namely, between the case the context is fully observed, and the case the context is unobserved. We refer to this class of decision problems as *LMDPs with prospective side information*. In such an environment an agent receives additional, weakly revealing, information on the latent context at the beginning of each episode. We show that, surprisingly, this problem is not captured by contemporary POMDP settings and is not solved by RL algorithms designed for partially observed environments. We then establish that any sample efficient algorithm must suffer at least $\Omega(K^{2/3})$-regret, as opposed to standard $\Omega(\sqrt{K})$ lower bounds. We design an algorithm with a matching upper bound that depends only polynomially on the problem parameters. This establishes exponential improvement in the sample complexity relative to the existing LMDP lower bound, when prospective information is not given in prior work.

In many real-world decision problems there is partially observed, hidden or latent information that remains fixed throughout an interaction. Such decision problems can be modeled as Latent Markov Decision Processes (LMDPs), where a latent variable is selected at the beginning of an interaction and is not disclosed to the agent initially. In last decade, there has been significant progress in designing learning algorithms for solving LMDPs under different structural assumptions. However, for general LMDPs, there is no known learning algorithm that provably matches the existing lower bound. We effectively resolve this open question, introducing the first sample-efficient algorithm for LMDPs without *any additional structural assumptions*. Our result builds off a new perspective on the role off-policy evaluation guarantees and coverage coefficient in LMDPs, a perspective, which has been overlooked in the context of exploration in partially observed environments. Specifically, we establish a novel off-policy evaluation lemma and introduce a new coverage coefficient for LMDPs. Then, we show how these can be used to derive near-optimal guarantees of an optimistic exploration algorithm. These results, we believe, can be valuable for a wide range of interactive learning problems beyond the LMDP class, and especially, for partially observed environments.

Embeddings play a pivotal role across various disciplines, offering compact representations of complex data structures. Randomized methods like Johnson-Lindenstrauss (JL) provide state-of-the-art and essentially unimprovable theoretical guarantees for achieving such representations. These guarantees are worst-case and in particular, neither the analysis, ${\textit{nor the algorithm}}$, takes into account any potential structural information of the data. The natural question is: must we randomize? Could we instead use an optimization-based approach, working directly with the data? A first answer is no: as we show, the distance-preserving objective of JL has a non-convex landscape over the space of projection matrices, with many bad stationary points. But this is not the final answer. We present a novel method motivated by diffusion models, that circumvents this fundamental challenge: rather than performing optimization directly over the space of projection matrices, we use optimization over the larger space of $\textit{random solution samplers}$, gradually reducing the variance of the sampler. We show that by moving through this larger space, our objective converges to a deterministic (zero variance) solution, avoiding bad stationary points. This method can also be seen as an optimization-based derandomization approach, and is an idea and method that we believe can be applied to many other problems.

We consider solving the low-rank matrix sensing problem with the Factorized Gradient Descent (FGD) method when the specified rank is larger than the true rank. We refer to this as over-parameterized matrix sensing. If the ground truth signal $\mathbf{X}^* \in \mathbb{R}^{d \times d}$ is of rank $r$, but we try to recover it using $\mathbf{F} \mathbf{F}^\top$ where $\mathbf{F} \in \mathbb{R}^{d \times k}$ and $k>r$, the existing statistical analysis either no longer holds or produces a vacuous statistical error upper bound (infinity) due to the flat local curvature of the loss function around the global maxima. By decomposing the factorized matrix $\mathbf{F}$ into separate column spaces to capture the impact of using $k > r$, we show that $\left\| {\mathbf{F}_t \mathbf{F}_t - \mathbf{X}^*} \right\|_F^2$ converges sub-linearly to a statistical error of $\tilde{\mathcal{O}} (k d \sigma^2/n)$ after $\tilde{\mathcal{O}}(\frac{\sigma_{r}}{\sigma}\sqrt{\frac{n}{d}})$ iterations, where $\mathbf{F}_t$ is the output of FGD after $t$ iterations, $\sigma^2$ is the variance of the observation noise, $\sigma_{r}$ is the $r$-th largest eigenvalue of $\mathbf{X}^*$, and $n$ is the number of samples. With a precise characterization of the convergence behavior and the statistical error, our results, therefore, offer a comprehensive picture of the statistical and computational complexity if we solve the over-parameterized matrix sensing problem with FGD.

Recent results established that EM enjoys global convergence for Gaussian Mixture Models. For Mixed Linear Regression, however, only local convergence results have been established, and those only for the high signal-to-noise ratio (SNR) regime. In this work, we completely characterize the global optimality of EM: we show that starting from any randomly initialized point, the EM algorithm converges to the true parameter${\beta }^{*}$at the minimax statistical rates under all SNR regimes. Toward this goal, we first show the global convergence of the EM algorithm at the population level. Then we provide a complete characterization of statistical and computational behaviors of EM under all SNR regimes with finite samples. In particular: (i) When the SNR is sufficiently large, the EM updates converge to the true parameter$ {\beta }^{*}$at the standard parametric convergence rate$O((d/n)^{1/2})$after$O(\log (n/d))$iterations. (ii) In the regime where the SNR is above$O((d/n)^{1/4})$and below some constant, the EM iterates converge to a$O({\mathrm { SNR}}^{-1} (d/n)^{1/2})$neighborhood of the true parameter, when the number of iterations is of the order$O({\mathrm { SNR}}^{-2} \log (n/d))$. (iii) In the low SNR regime where the SNR is below$O((d/n)^{1/4})$, we show that EM converges to a$O((d/n)^{1/4})$neighborhood of the true parameters, after$O((n/d)^{1/2})$iterations. By providing tight convergence guarantees of the EM algorithm in middle-to-low SNR regimes, we reveal that in low SNR, EM changes rate, matching the$n^{-1/4}$rate of the MLE, a behavior that previous work had been unable to show.

This work considers the problem of finding a first-order stationary point of a non-convex function with potentially unbounded smoothness constant using a stochastic gradient oracle. We focus on the class of $(L_0,L_1)$-smooth functions proposed by Zhang et al. (ICLR’20). Empirical evidence suggests that these functions more closely capture practical machine learning problems as compared to the pervasive $L_0$-smoothness. This class is rich enough to include highly non-smooth functions, such as $\exp(L_1 x)$ which is $(0,\mathcal{O}(L_1))$-smooth. Despite the richness, an emerging line of works achieves the $\widetilde{\mathcal{O}}(\frac{1}{\sqrt{T}})$ rate of convergence when the noise of the stochastic gradients is deterministically and uniformly bounded. This noise restriction is not required in the $\L_0$-smooth setting, and in many practical settings is either not satisfied, or results in weaker convergence rates with respect to the noise scaling of the convergence rate.We develop a technique that allows us to prove $\mathcal{O}(\frac{\mathrm{poly}\log(T)}{\sqrt{T}})$ convergence rates for $(L_0,L_1)$-smooth functions without assuming uniform bounds on the noise support. The key innovation behind our results is a carefully constructed stopping time $\tau$ which is simultaneously “large” on average, yet also allows us to treat the adaptive step sizes before $\tau$ as (roughly) independent of the gradients. For general $(L_0,L_1)$-smooth functions, our analysis requires the mild restriction that the multiplicative noise parameter $\sigma_1 < 1$. For a broad subclass of $(L_0,L_1)$-smooth functions, our convergence rate continues to hold when $\sigma_1 \geq 1$. By contrast, we prove that many algorithms analyzed by prior works on $(L_0,L_1)$-smooth optimization diverge with constant probability even for smooth and strongly-convex functions when $\sigma_1 > 1$.

We consider episodic reinforcement learning in reward-mixing Markov decision processes (RMMDPs): at the beginning of every episode nature randomly picks a latent reward model among $M$ candidates and an agent interacts with the MDP throughout the episode for $H$ time steps. Our goal is to learn a near-optimal policy that nearly maximizes the $H$ time-step cumulative rewards in such a model. Prior work established an upper bound for RMMDPs with $M=2$. In this work, we resolve several open questions for the general RMMDP setting. We consider an arbitrary $M\ge2$ and provide a sample-efficient algorithm–$EM^2$–that outputs an $\epsilon$-optimal policy using $O \left(\epsilon^{-2} \cdot S^d A^d \cdot \text{poly}(H, Z)^d \right)$ episodes, where $S, A$ are the number of states and actions respectively, $H$ is the time-horizon, $Z$ is the support size of reward distributions and $d=O(\min(M,H))$. We also provide a $(SA)^{\Omega(\sqrt{M})} / \epsilon^{2}$ lower bound, supporting that super-polynomial sample complexity in $M$ is necessary.

Deep Neural Networks and Reinforcement Learning methods have empirically shown great promise in tackling challenging combinatorial problems. In those methods a deep neural network is used as a solution generator which is then trained by gradient-based methods (e.g., policy gradient) to successively obtain better solution distributions. In this work we introduce a novel theoretical framework for analyzing the effectiveness of such methods. We ask whether there exist generative models that (i) are expressive enough to generate approximately optimal solutions; (ii) have a tractable, i.e, polynomial in the size of the input, number of parameters; (iii) their optimization landscape is benign in the sense that it does not contain sub-optimal stationary points. Our main contribution is a positive answer to this question. Our result holds for a broad class of combinatorial problems including Max- and Min-Cut, Max-$k$-CSP, Maximum-Weight-Bipartite-Matching, and the Traveling Salesman Problem. As a byproduct of our analysis we introduce a novel regularization process over vanilla gradient descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.

We present the first framework to solve linear inverse problems leveraging pre-trained \textit{latent} diffusion models. Previously proposed algorithms (such as DPS and DDRM) only apply to \textit{pixel-space} diffusion models. We theoretically analyze our algorithm showing provable sample recovery in a linear model setting. The algorithmic insight obtained from our analysis extends to more general settings often considered in practice. Experimentally, we outperform previously proposed posterior sampling algorithms in a wide variety of problems including random inpainting, block inpainting, denoising, deblurring, destriping, and super-resolution.

We study the problem of learning tree-structured Markov random fields (MRF) on discrete random variables with common support when the observations are corrupted by a k-ary symmetric noise channel with unknown probability of error. For Ising models (support size = 2), past work has shown that graph structure can only be recovered up to the leaf clusters (a leaf node, its parent, and its siblings form a leaf cluster) and exact recovery is impossible. No prior work has addressed the setting of support size of 3 or more, and indeed this setting is far richer. As we show, when the support size is 3 or more, the structure of the leaf clusters may be partially or fully identifiable. We provide a precise characterization of this phenomenon and show that the extent of recoverability is dictated by the joint PMF of the random variables. In particular, we provide necessary and sufficient conditions for exact recoverability. Furthermore, we present a polynomial time, sample efficient algorithm that recovers the exact tree when this is possible, or up to the unidentifiability as promised by our characterization, when full recoverability is impossible. Finally, we demonstrate the efficacy of our algorithm experimentally.

We study convergence rates of AdaGrad-Norm as an exemplar of adaptive stochastic gradient methods (SGD), where the step sizes change based on observed stochastic gradients, for minimizing non-convex, smooth objectives. Despite their popularity, the analysis of adaptive SGD lags behind that of non adaptive methods in this setting. Specifically, all prior works rely on some subset of the following assumptions: (i) uniformly-bounded gradient norms, (ii) uniformly-bounded stochastic gradient variance (or even noise support), (iii) conditional independence between the step size and stochastic gradient. In this work, we show that AdaGrad-Norm exhibits an order optimal convergence rate of $\mathcal{O}\left(\frac{\mathrm{poly}\log(T)}{\sqrt{T}}\right)$ after $T$ iterations under the same assumptions as optimally-tuned non adaptive SGD (unbounded gradient norms and affine noise variance scaling), and crucially, without needing any tuning parameters. We thus establish that adaptive gradient methods exhibit order-optimal convergence in much broader regimes than previously understood.

We study the problem of Gaussian bandits with general side information, as first introduced by Wu, Szepesvári, and György. In this setting, the play of an arm reveals information about other arms, according to an arbitrary a priori known side information matrix: each element of this matrix encodes the fidelity of the information that the “row" arm reveals about the “column" arm. In the case of Gaussian noise, this model subsumes standard bandits, full-feedback, and graph-structured feedback as special cases. In this work, we first construct an LP-based asymptotic instance-dependent lower bound on the regret. The LP optimizes the cost (regret) required to reliably estimate the suboptimality gap of each arm. This LP lower bound motivates our main contribution: the first known asymptotically optimal algorithm for this general setting.

Motivated by online recommendation systems, we propose the problem of finding the optimal policy in multitask contextual bandits when a small fraction $\alpha < 1/2$ of tasks (users) are arbitrary and adversarial. The remaining fraction of good users share the same instance of contextual bandits with $S$ contexts and $A$ actions (items). Naturally, whether a user is good or adversarial is not known in advance. The goal is to robustly learn the policy that maximizes rewards for good users with as few user interactions as possible. Without adversarial users, established results in collaborative filtering show that $O(1/\epsilon^2)$ per-user interactions suffice to learn a good policy, precisely because information can be shared across users. This parallelization gain is fundamentally altered by the presence of adversarial users: unless there are super-polynomial number of users, we show a lower bound of $\tilde{\Omega}(\min(S,A) \cdot \alpha^2 / \epsilon^2)$ per-user interactions to learn an $\epsilon$-optimal policy for the good users. We then show we can achieve an $\tilde{O}(\min(S,A)\cdot \alpha/\epsilon^2)$ upper-bound, by employing efficient robust mean estimators for both uni-variate and high-dimensional random variables. We also show that this can be improved depending on the distributions of contexts.

We consider a multi-armed bandit problem with $M$ latent contexts, where an agent interacts with the environment for an episode of $H$ time steps. Depending on the length of the episode, the learner may not be able to estimate accurately the latent context. The resulting partial observation of the environment makes the learning task significantly more challenging. Without any additional structural assumptions, existing techniques to tackle partially observed settings imply the decision maker can learn a near-optimal policy with $O(A)^H$ episodes, but do not promise more. In this work, we show that learning with {\em polynomial} samples in $A$ is possible. We achieve this by using techniques from experiment design. Then, through a method-of-moments approach, we design a procedure that provably learns a near-optimal policy with $O(\poly(A) + \poly(M,H)^{\min(M,H)})$ interactions. In practice, we show that we can formulate the moment-matching via maximum likelihood estimation. In our experiments, this significantly outperforms the worst-case guarantees, as well as existing practical methods.

The stochastic multi-armed bandit setting has been recently studied in the non-stationary regime, where the mean payoff of each action is a non-decreasing function of the number of rounds passed since it was last played. This model captures natural behavioral aspects of the users which crucially determine the performance of recommendation platforms, ad placement systems, and more. Even assuming prior knowledge of the mean payoff functions, computing an optimal planning in the above model is NP-hard, while the state-of-the-art is a $1/4$-approximation algorithm for the case where at most one arm can be played per round. We first focus on the setting where the mean payoff functions are known. In this setting, we significantly improve the best-known guarantees for the planning problem by developing a polynomial-time $(1-{1}/{e})$-approximation algorithm (asymptotically and in expectation), based on a novel combination of randomized LP rounding and a time-correlated (interleaved) scheduling method. Furthermore, our algorithm achieves improved guarantees -- compared to prior work -- for the case where more than one arms can be played at each round. Moving to the bandit setting, when the mean payoff functions are initially unknown, we show how our algorithm can be transformed into a bandit algorithm with sublinear regret.

We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.

We study a novel variant of the multi-armed bandit problem, where at each time step, the player observes an independently sampled context that determines the arms’ mean rewards. However, playing an arm blocks it (across all contexts) for a fixed number of future time steps. The above contextual setting captures important scenarios such as recommendation systems or ad placement with diverse users. This problem has been recently studied [Dickerson et al., AAAI 2018] in the full-information setting (i.e., assuming knowledge of the mean context-dependent arm rewards), where competitive ratio bounds have been derived. We focus on the bandit setting, where these means are initially unknown; we propose a UCB-based variant of the full-information algorithm that guarantees a $\mathcal{O}(\log T)$-regret w.r.t. an $\alpha$-optimal strategy in $T$ time steps, matching the $\Omega(\log(T))$ regret lower bound in this setting. Due to the time correlations caused by blocking, existing techniques for upper bounding regret fail. For proving our regret bounds, we introduce the novel concepts of delayed exploitation and opportunistic subsampling and combine them with ideas from combinatorial bandits and non-stationary Markov chains coupling.

We study the convergence rates of the EM algorithm for learning two-component mixed linear regression under all regimes of signal-to-noise ratio (SNR). We resolve a long-standing question that many recent results have attempted to tackle: we completely characterize the convergence behavior of EM, and show that the EM algorithm achieves minimax optimal sample complexity under all SNR regimes. In particular, when the SNR is sufficiently large, the EM updates converge to the true parameter $\theta^{*}$ at the standard parametric convergence rate $\calo((d/n)^{1/2})$ after $\calo(\log(n/d))$ iterations. In the regime where the SNR is above $\calo((d/n)^{1/4})$ and below some constant, the EM iterates converge to a $\calo({\rm SNR}^{-1} (d/n)^{1/2})$ neighborhood of the true parameter, when the number of iterations is of the order $\calo({\rm SNR}^{-2} \log(n/d))$. In the low SNR regime where the SNR is below $\calo((d/n)^{1/4})$, we show that EM converges to a $\calo((d/n)^{1/4})$ neighborhood of the true parameters, after $\calo((n/d)^{1/2})$ iterations. Notably, these results are achieved under mild conditions of either random initialization or an efficiently computable local initialization. By providing tight convergence guarantees of the EM algorithm in middle-to-low SNR regimes, we fill the remaining gap in the literature, and significantly, reveal that in low SNR, EM changes rate, matching the $n^{-1/4}$ rate of the MLE, a behavior that previous work had been unable to show.

Recent work has considered natural variations of the {\em multi-armed bandit} problem, where the reward distribution of each arm is a special function of the time passed since its last pulling. In this direction, a simple (yet widely applicable) model is that of {\em blocking bandits}, where an arm becomes unavailable for a deterministic number of rounds after each play. In this work, we extend the above model in two directions: (i) We consider the general combinatorial setting where more than one arms can be played at each round, subject to feasibility constraints. (ii) We allow the blocking time of each arm to be stochastic. We first study the computational/unconditional hardness of the above setting and identify the necessary conditions for the problem to become tractable (even in an approximate sense). Based on these conditions, we provide a tight analysis of the approximation guarantee of a natural greedy heuristic that always plays the maximum expected reward feasible subset among the available (non-blocked) arms. When the arms’ expected rewards are unknown, we adapt the above heuristic into a bandit algorithm, based on UCB, for which we provide sublinear (approximate) regret guarantees, matching the theoretical lower bounds in the limiting case of absence of delays.

Learning a near optimal policy in a partially observable system remains an elusive challenge in contemporary reinforcement learning. In this work, we consider episodic reinforcement learning in a reward-mixing Markov decision process (MDP). There, a reward function is drawn from one of $M$ possible reward models at the beginning of every episode, but the identity of the chosen reward model is not revealed to the agent. Hence, the latent state space, for which the dynamics are Markovian, is not given to the agent. We study the problem of learning a near optimal policy for two reward-mixing MDPs. Unlike existing approaches that rely on strong assumptions on the dynamics, we make no assumptions and study the problem in full generality. Indeed, with no further assumptions, even for two switching reward-models, the problem requires several new ideas beyond existing algorithmic and analysis techniques for efficient exploration. We provide the first polynomial-time algorithm that finds an $\epsilon$-optimal policy after exploring $\tilde{O}(poly(H,\epsilon^{-1}) \cdot S^2 A^2)$ episodes, where $H$ is time-horizon and $S, A$ are the number of states and actions respectively. This is the first efficient algorithm that does not require any assumptions in partially observed environments where the observation space is smaller than the latent state space.

In this work, we consider the regret minimization problem for reinforcement learning in latent Markov Decision Processes (LMDP). In an LMDP, an MDP is randomly drawn from a set of $M$ possible MDPs at the beginning of the interaction, but the identity of the chosen MDP is not revealed to the agent. We first show that a general instance of LMDPs requires at least $\Omega((SA)^M)$ episodes to even approximate the optimal policy. Then, we consider sufficient assumptions under which learning good policies requires polynomial number of episodes. We show that the key link is a notion of separation between the MDP system dynamics. With sufficient separation, we provide an efficient algorithm with local guarantee, {\it i.e.,} providing a sublinear regret guarantee when we are given a good initialization. Finally, if we are given standard statistical sufficiency assumptions common in the Predictive State Representation (PSR) literature (e.g., \cite{boots2011online}) and a reachability assumption, we show that the need for initialization can be removed.

A recent line of research focuses on the study of stochastic multi-armed bandits (MAB), in the case where temporal correlations of specific structure are imposed between the player's actions and the reward distributions of the arms. These correlations lead to (sub-)optimal solutions that exhibit interesting dynamical patterns -- a phenomenon that yields new challenges both from an algorithmic as well as a learning perspective. In this work, we extend the above direction to a combinatorial semi-bandit setting and study a variant of stochastic MAB, where arms are subject to matroid constraints and each arm becomes unavailable (blocked) for a fixed number of rounds after each play. A natural common generalization of the state-of-the-art for blocking bandits, and that for matroid bandits, only guarantees a $\frac{1}{2}$-approximation for general matroids. In this paper we develop the novel technique of correlated (interleaved) scheduling, which allows us to obtain a polynomial-time $(1 - \frac{1}{e})$-approximation algorithm (asymptotically and in expectation) for any matroid. Along the way, we discover an interesting connection to a variant of Submodular Welfare Maximization, for which we provide (asymptotically) matching upper and lower approximability bounds. In the case where the mean arm rewards are unknown, our technique naturally decouples the scheduling from the learning problem, and thus allows to control the $(1-\frac{1}{e})$-approximate regret of a UCB-based adaptation of our online algorithm.

We study the convergence of the Expectation-Maximization (EM) algorithm for mixtures of linear regressions with an arbitrary number $k$ of components. We show that as long as signal-to-noise ratio (SNR) is $\tilde{\Omega}(k)$, well-initialized EM converges to the true regression parameters. Previous results for $k \geq 3$ have only established local convergence for the noiseless setting, i.e., where SNR is infinitely large. Our results enlarge the scope to the environment with noises, and notably, we establish a statistical error rate that is independent of the norm (or pairwise distance) of the regression parameters. In particular, our results imply exact recovery as $\sigma \rightarrow 0$, in contrast to most previous local convergence results for EM, where the statistical error scaled with the norm of parameters. Standard moment-method approaches may be applied to guarantee we are in the region where our local convergence guarantees apply.

We provide a novel – and to the best of our knowledge, the first – algorithm for high dimensional sparse regression with constant fraction of corruptions in explanatory and/or response variables. Our algorithm recovers the true sparse parameters with sub-linear sample complexity,in the presence of a constant fraction of arbitrary corruptions. Our main contribution is a robust variant of Iterative Hard Thresholding. Using this, we provide accurate estimators:when the covariance matrix in sparse regression is identity, our error guarantee is near information-theoretically optimal. We then deal with robust sparse regression with unknown structured covariance matrix. We propose a filtering algorithm whichconsists of a novel randomized outlier removal technique for robust sparse mean estimation that may be of interest in its own right: the filtering algorithm is flexible enough to deal with unknown covariance.Also, it is orderwise more efficient computationally than the ellipsoid algorithm.Using sub-linear sample complexity, our algorithm achieves the best known (and first) error guarantee. We demonstrate the effectiveness on large-scale sparse regression problems with arbitrary corruptions.

Large-scale machine learning training suffers from two prior challenges, specifically for nuclear-norm constrained problems with distributed systems: the synchronization slowdown due to the straggling workers, and high communication costs. In this work, we propose an asynchronous Stochastic Frank Wolfe (SFW-asyn) method, which, for the first time, solves the two problems simultaneously, while successfully maintaining the same convergence rate as the vanilla SFW. We implement our algorithm in python (with MPI) to run on Amazon EC2, and demonstrate that SFW-asyn yields speed-ups almost linear to the number of machines compared to the vanilla SFW.

We consider the problem of spherical Gaussian Mixture models with $k \geq 3$ components when the components are well separated. A fundamental previous result established that separation of $\Omega(\sqrt{\log k})$ is necessary and sufficient for identifiability of the parameters with \textit{polynomial} sample complexity (Regev and Vijayaraghavan, 2017). In the same context, we show that $\tilde{O} (kd/\epsilon^2)$ samples suffice for any $\epsilon \lesssim 1/k$, closing the gap from polynomial to linear, and thus giving the first optimal sample upper bound for the parameter estimation of well-separated Gaussian mixtures. We accomplish this by proving a new result for the Expectation-Maximization (EM) algorithm: we show that EM converges locally, under separation $\Omega(\sqrt{\log k})$. The previous best-known guarantee required $\Omega(\sqrt{k})$ separation (Yan, et al., 2017). Unlike prior work, our results do not assume or use prior knowledge of the (potentially different) mixing weights or variances of the Gaussian components. Furthermore, our results show that the finite-sample error of EM does not depend on non-universal quantities such as pairwise distances between means of Gaussian components.

We consider the problem of learning the weighted edges of a balanced mixture of two undirected graphs from epidemic cascades. While mixture models are popular modeling tools, algorithmic development with rigorous guarantees has lagged. Graph mixtures are apparently no exception: until now, very little is known about whether this problem is solvable. To the best of our knowledge, we establish the first necessary and sufficient conditions for this problem to be solvable in polynomial time on edge-separated graphs. When the conditions are met, i.e., when the graphs are connected with at least three edges, we give an efficient algorithm for learning the weights of both graphs with optimal sample complexity (up to log factors). We give complementary results and provide sample-optimal (up to log factors) algorithms for mixtures of directed graphs of out-degree at least three, and for mixture of undirected graphs of unbalanced and/or unknown priors.

We consider a covariate shift problem where one has access to several different training datasets for the same learning problem and a small validation set which possibly differs from all the individual training distributions. The distribution shift is due, in part, to \emph{unobserved} features in the datasets. The objective, then, is to find the best mixture distribution over the training datasets (with only observed features) such that training a learning algorithm using this mixture has the best validation performance. Our proposed algorithm, \textsf{Mix&amp;Match}, combines stochastic gradient descent (SGD) with optimistic tree search and model re-use (evolving partially trained models with samples from different mixture distributions) over the space of mixtures, for this task. We prove a novel high probability bound on the final SGD iterate without relying on a global gradient norm bound, and use it to show the advantages of model re-use. Additionally, we provide simple regret guarantees for our algorithm with respect to recovering the optimal mixture, given a total budget of SGD evaluations. Finally, we validate our algorithm on two real-world datasets.

We consider estimating a high dimensional signal in $\R^n$ using a sublinear number of linear measurements. In analogy to classical compressed sensing, here we assume a generative model as a prior, that is, we assume the signal is represented by a deep generative model $G: \R^k \rightarrow \R^n$. Classical recovery approaches such as empirical risk minimization (ERM) are guaranteed to succeed when the measurement matrix is sub-Gaussian. However, when the measurement matrix and measurements are heavy tailed or have outliers, recovery may fail dramatically. In this paper we propose an algorithm inspired by the Median-of-Means (MOM). Our algorithm guarantees recovery for heavy tailed data, even in the presence of outliers. Theoretically, our results show our novel MOM-based algorithm enjoys the same sample complexity guarantees as ERM under sub-Gaussian assumptions. Our experiments validate both aspects of our claims: other algorithms are indeed fragile and fail under heavy tailed and/or corrupted data, while our approach exhibits the predicted robustness.

We study the problem of discovering the simplest latent variable that can make two observed discrete variables conditionally independent. The minimum entropy required for such a latent is known as common entropy in information theory. We extend this notion to Renyi common entropy by minimizing the Renyi entropy of the latent variable. To efficiently compute common entropy, we propose an iterative algorithm that can be used to discover the trade-off between the entropy of the latent variable and the conditional mutual information of the observed variables. We show two applications of common entropy in causal inference: First, under the assumption that there are no low-entropy mediators, it can be used to distinguish direct causation from spurious correlation among almost all joint distributions on simple causal graphs with two observed variables. Second, common entropy can be used to improve constraint-based methods such as PC or FCI algorithms in the small-sample regime, where these methods are known to struggle. We propose a modification to these constraint-based methods to assess if a separating set found by these algorithms are valid using common entropy. We finally evaluate our algorithms on synthetic and real data to establish their performance.

In this paper we study the problem of escaping from saddle points and achieving second-order optimality in a decentralized setting where a group of agents collaborate to minimize their aggregate objective function. We provide a non-asymptotic (finite-time) analysis and show that by following the idea of perturbed gradient descent, it is possible to converge to a second-order stationary point in a number of iterations which depends linearly on dimension and polynomially on the accuracy of second-order stationary point. Doing this in a communication-efficient manner requires overcoming several challenges, from identifying (first order) stationary points in a distributed manner, to adapting the perturbed gradient framework without prohibitive communication complexity. Our proposed Perturbed Decentralized Gradient Tracking (PDGT) method consists of two major stages: (i) a gradient-based step to find a first-order stationary point and (ii) a perturbed gradient descent step to escape from a first-order stationary point, if it is a saddle point with sufficient curvature. As a side benefit of our result, in the case that all saddle points are non-degenerate (strict), the proposed PDGT method finds a local minimum of the considered decentralized optimization problem in a finite number of iterations.

The Expectation-Maximization algorithm is perhaps the most broadly used algorithm for inference of latent variable problems. A theoretical understanding of its performance, however, largely remains lacking. Recent results established that EM enjoys global convergence for Gaussian Mixture Models. For Mixed Linear Regression, however, only local convergence results have been established, and those only for the high SNR regime. We show here that EM converges for mixed linear regression with two components (it is known that it may fail to converge for three or more), and moreover that this convergence holds for random initialization. Our analysis reveals that EM exhibits very different behavior in Mixed Linear Regression from its behavior in Gaussian Mixture Models, and hence our proofs require the development of several new ideas.

Consider jointly Gaussian random variables whose conditional independence structure is specified by a graphical model. If we observe realizations of the variables, we can compute the covariance matrix, and it is well known that the support of the inverse covariance matrix corresponds to the edges of the graphical model. Instead, suppose we only have noisy observations. If the noise at each node is independent, we can compute the sum of the covariance matrix and an unknown diagonal. The inverse of this sum is (in general) dense. We ask: can the original independence structure be recovered? We address this question for tree structured graphical models. We prove that this problem is unidentifiable, but show that this unidentifiability is limited to a small class of candidate trees. We further present additional constraints under which the problem is identifiable. Finally, we provide an O(n^3) algorithm to find this equivalence class of trees.

We propose a generalized variant of Frank-Wolfe algorithm for solving a class of sparse/low-rank optimization problems. Our formulation includes Elastic Net, regularized SVMs and phase retrieval as special cases. The proposed Primal-Dual Block Generalized Frank-Wolfe algorithm reduces the per-iteration cost while maintaining linear convergence rate. The per iteration cost of our method depends on the structural complexity of the solution (i.e. sparsity/low-rank) instead of the ambient dimension. We empirically show that our algorithm outperforms the state-of-the-art methods on (multi-class) classification tasks.

We present a novel method for frequentist statistical inference in M-estimation problems, based on stochastic gradient descent (SGD) with a fixed step size: we demonstrate that the average of such SGD sequences can be used for statistical inference, after proper scaling. An intuitive analysis using the Ornstein-Uhlenbeck process suggests that such averages are asymptotically normal. To show the merits of our scheme, we apply it to both synthetic and real data sets, and demonstrate that its accuracy is comparable to classical statistical methods, while requiring potentially far less computation.

The univariate statistics of bandpass-filtered images provide powerful features that drive many successful image quality assessment (IQA) algorithms. Bivariate Natural Scene Statistics (NSS), which model the joint statistics of multiple bandpass image samples also provide potentially powerful features to assess the perceptual quality of images, by capturing both image and distortion correlations. Here, we make the first attempt to use bivariate NSS features to build a model of no-reference image quality prediction. We show that our bivariate model outperforms existing state of the art image quality predictors.

A rank-$r$ matrix $X \in \mathbb{R}^{m \times n}$ can be written as a product $U V^\top$, where $U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{n \times r}$. One could exploit this observation in optimization: e.g., consider the minimization of a convex function $f(X)$ over rank-$r$ matrices, where the set of low-rank matrices is modeled via $UV^\top$. Though such parameterization reduces the number of variables and is more computationally efficient (of particular interest is the case $r \ll \min\{m, n\}$), it comes at a cost: $f(U V^\top)$ becomes a nonconvex function w.r.t. $U$ and $V$. We study such parameterization on generic convex objectives $f$ and focus on first-order, gradient descent algorithms. We propose the bifactored gradient descent (\textttBFGD) algorithm, an efficient first-order method that operates directly on the $U, V$ factors. We show that when $f$ is (restricted) smooth, \textttBFGD has local sublinear convergence; when $f$ is both (restricted) smooth and (restricted) strongly convex, it has local linear convergence. For several applications, we provide simple and efficient initialization schemes that provide initial conditions, good enough for the above convergence results to hold, globally. Extensive experimental results support our arguments that \textttBFGD is an efficient and accurate nonconvex method, compared to state-of-the-art approaches.

Previous work on natural scene statistics (NSS)-based image models has focused primarily on characterizing the univariate bandpass statistics of single pixels. These models have proven to be powerful tools driving a variety of computer vision and image/video processing applications, including depth estimation, image quality assessment, and image denoising, among others. Multivariate NSS models descriptive of the joint distributions of spatially separated bandpass image samples have, however, received relatively little attention. Here, we develop a closed form bivariate spatial correlation model of bandpass and normalized image samples that completes an existing 2D joint generalized Gaussian distribution model of adjacent bandpass pixels. Our model is built using a set of diverse, high-quality naturalistic photographs, and as a control, we study the model properties on white noise. We also study the way the model fits are affected when the images are modified by common distortions.

We consider the mixed regression problem with two components, under adversarial and stochastic noise. We give a convex optimization formulation that provably recovers the true solution, as well as a nonconvex formulation that works under more general settings and remains tractable. Upper bounds are provided on the recovery errors for both arbitrary noise and stochastic noise models. We also give matching minimax lower bounds (up to log factors), showing that our algorithm is information-theoretical optimal in a precise sense. Our results represent the first tractable algorithm guaranteeing successful recovery with tight bounds on recovery errors and sample complexity. Moreover, we pinpoint the statistical cost of mixtures: our minimax-optimal results indicate that the mixture poses a fundamentally more difficult problem in the low-SNR regime, where the learning rate changes.

We present minimax bounds for classification and clustering error in the setting where covariates are drawn from a mixture of two isotropic Gaussian distributions. Here, we define clustering error in a discriminative fashion, demonstrating fundamental connections between classification (supervised) and clustering (unsupervised). For both classification and clustering, our lower bounds show that without enough samples, the best any classifier or clustering rule can do is close to random guessing. For classification, as part of our upper bound analysis, we show that Fisher's linear discriminant achieves a fast minimax rate $\Theta(1/n)$ with enough samples $n$. For clustering, as part of our upper bound analysis, we show that a clustering rule constructed using principal component analysis achieves the minimax rate with enough samples. We also provide lower and upper bounds for the high-dimensional sparse setting where the dimensionality of the covariates $p$ is potentially larger than the number of samples $n$, but where the difference between the Gaussian means is sparse.

We consider the non-square matrix sensing problem, under restricted isometry property (RIP) assumptions. We focus on the non-convex formulation, where any rank-r matrix $X ∈R^m x n$ is represented as $UV^T$, where $U ∈R^m x r$ and $V ∈R^n x r$. In this paper, we complement recent findings on the non-convex geometry of the analogous PSD setting [5], and show that matrix factorization does not introduce any spurious local minima, under RIP.

Energy-efficient processing of large matrices for big-data applications using hardware acceleration is an intense area of research. Sketching of large matrices into their lower-dimensional representations is an effective strategy. For the first time, this paper develops a highly energy-efficient hardware implementation of a class of sketching methods based on random projections, known as Johnson Lindenstrauss (JL) transform. Crucially, we show how the randomness inherent in the projection matrix can be exploited to create highly efficient fixed-point arithmetic realizations of several machine-learning applications. Specifically, the transform's random matrices have two key properties that allow for significant energy gains in hardware implementations. The first is the smoothing property that allows us to drastically reduce operand bit-width in computation of the JL transform itself. The second is the randomizing property that allows bit-width reduction in subsequent machine-learning applications. Further, we identify a random matrix construction method that exploits the special sparsity structure to result in the most hardware-efficient realization and implement the highly optimized transform on an FPGA. Experimental results on (1) the k-nearest neighbor (KNN) classification and (2) the principal component analysis (PCA) show that with the same bit-width the proposed flow utilizing random projection achieves an up to 7× improvement in both latency and energy. Furthermore, by exploiting the smoothing and randomizing properties we are able to use a 1-bit instead of a 4-bit multiplier within KNN, which results in additional 50% and 6% improvement in area and energy respectively. The proposed I/O streaming strategy along with the hardware-efficient JL algorithm identified by us is able to achieve a 50% runtime reduction, a 17% area reduction in the stage of random projection compared to a standard design.

We consider the problem of Robust PCA in the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem. From a statistical standpoint this problem has been recently well-studied, and conditions on when recovery is possible (how many observations do we need, how many corruptions can we tolerate) via polynomial-time algorithms is by now understood. This paper presents and analyzes a non-convex optimization approach that greatly reduces the computational complexity of the above problems, compared to the best available algorithms. In particular, in the fully observed case, with $r$ denoting rank and $d$ dimension, we reduce the complexity from $O(r^2d^2\log(1/\epsilon))$ to $O(rd^2\log(1/\epsilon))$ -- a big savings when the rank is big. For the partially observed case, we show the complexity of our algorithm is no more than $O(r^4d\log(d)\log(1/\epsilon))$. Not only is this the best-known run-time for a provable algorithm under partial observation, but in the setting where $r$ is small compared to $d$, it also allows for near-linear-in-$d$ run-time that can be exploited in the fully-observed case as well, by simply running our algorithm on a subset of the observations.

We consider the weakly supervised binary classification problem where the labels are randomly flipped with probability $1-\alpha$. Although there exist numerous algorithms for this problem, it remains theoretically unexplored how the statistical accuracies and computational efficiency of these algorithms depend on the degree of supervision, which is quantified by $\alpha$. In this paper, we characterize the effect of $\alpha$ by establishing the information-theoretic and computational boundaries, namely, the minimax-optimal statistical accuracy that can be achieved by all algorithms, and polynomial-time algorithms under an oracle computational model. For small $\alpha$, our result shows a gap between these two boundaries, which represents the computational price of achieving the information-theoretic boundary due to the lack of supervision. Interestingly, we also show that this gap narrows as $\alpha$ increases. In other words, having more supervision, i.e., more correct labels, not only improves the optimal statistical accuracy as expected, but also enhances the computational efficiency for achieving such accuracy.

Two key traits of 5G cellular networks are much higher base station (BS) densities-especially in the case of low-power BSs-and the use of massive MIMO at these BSs. This paper explores how massive MIMO can be used to jointly maximize the offloading gains and minimize the interference challenges arising from adding small cells. We consider two interference management approaches: joint transmission (JT) with local precoding, where users are served simultaneously by multiple BSs without requiring channel state information exchanges among cooperating BSs, and resource blanking, where some macro BS resources are left blank to reduce the interference in the small cell downlink. A key advantage offered by massive MIMO is channel hardening, which enables to predict instantaneous rates a priori. This allows us to develop a unified framework, where resource allocation is cast as a network utility maximization (NUM) problem, and to demonstrate large gains in cell-edge rates based on the NUM solution. We propose an efficient dual subgradient based algorithm, which converges towards the NUM solution. A scheduling scheme is also proposed to approach the NUM solution. Simulations illustrate more than 2x rate gain for 10th percentile users vs. an optimal association without interference management.

This paper considers the problem of matrix completion when some number of the columns are completely and arbitrarily corrupted, potentially by a malicious adversary. It is well known that standard algorithms for matrix completion can return arbitrarily poor results, if even a single column is corrupted. One direct application comes from robust collaborative filtering. Here, some number of users are so-called manipulators who try to skew the predictions of the algorithm by calibrating their inputs to the system. In this paper, we develop an efficient algorithm for this problem based on a combination of a trimming procedure and a convex program that minimizes the nuclear norm and the ℓ1,2norm. Our theoretical results show that given a vanishing fraction of observed entries, it is nevertheless possible to complete the underlying matrix even when the number of corrupted columns grows. Significantly, our results hold without any assumptions on the locations or values of the observed entries of the manipulated columns. Moreover, we show by an information-theoretic argument that our guarantees are nearly optimal in terms of the fraction of sampled entries on the authentic columns, the fraction of corrupted columns, and the rank of the underlying matrix. Our results therefore sharply characterize the tradeoffs between sample, robustness, and rank in matrix completion.

Model order reduction exploiting the spectral properties of the admittance matrix, known as the graph Laplacian, to control the approximation accuracy is a promising new class of approaches to power grid analysis. In this paper we introduce a method that allows a dramatic increase in the resulting graph sparsity and can handle large dense input graphs. The method is based on the observation that the information about the realistic ranges of port currents can be used to significantly improve the resulting graph sparsity. In practice, port currents cannot vary unboundedly and the estimates of peak currents are often available early in the design cycle. However, the existing methods including the sampling-based spectral sparsification approach [11] cannot utilize this information.

Binary embedding is a nonlinear dimension reduction methodology where high dimensional data are embedded into the Hamming cube while preserving the structure of the original space. Specifically, for an arbitrary N distinct points in \mathbbS^p-1, our goal is to encode each point using m-dimensional binary strings such that we can reconstruct their geodesic distance up to δuniform distortion. Existing binary embedding algorithms either lack theoretical guarantees or suffer from running time O(mp). We make three contributions: (1) we establish a lower bound that shows any binary embedding oblivious to the set of points requires m =Ω(\frac1δ^2\logN) bits and a similar lower bound for non-oblivious embeddings into Hamming distance; (2) we propose a novel fast binary embedding algorithm with provably optimal bit complexity m = O(\frac1 δ^2\logN) and near linear running time O(p \log p) whenever \log N ≪δ\sqrtp, with a slightly worse running time for larger \log N; (3) we also provide an analytic result about embedding a general set of points K ⊆\mathbbS^p-1 with even infinite size. Our theoretical findings are supported through experiments on both synthetic and real data sets.

In many networks the operator is faced with nodes that report a potentially important phenomenon such as failures, illnesses, and viruses. The operator is faced with the question: Is it spreading over the network, or simply occurring at random? We seek to answer this question from highly noisy and incomplete data, where at a single point in time we are given a possibly very noisy subset of the infected population (including false positives and negatives). While previous work has focused on uniform spreading rates for the infection, heterogeneous graphs with unequal edge weights are more faithful models of reality. Critically, the network structure may not be fully known and modeling epidemic spread on unknown graphs relies on non-homogeneous edge (spreading) weights. Such heterogeneous graphs pose considerable challenges, requiring both algorithmic and analytical development. We develop an algorithm that can distinguish between a spreading phenomenon and a randomly occurring phenomenon while using only local information and not knowing the complete network topology and the weights. Further, we show that this algorithm can succeed even in the presence of noise, false positives and unknown graph edges.

Latent models are a fundamental modeling tool in machine learning applications, but they present significant computational and analytical challenges. The popular EM algorithm and its variants, is a much used algorithmic tool; yet our rigorous understanding of its performance is highly incomplete. Recently, work in [1] has demonstrated that for an important class of problems, EM exhibits linear local convergence. In the high-dimensional setting, however, the M-step may not be well defined. We address precisely this setting through a unified treatment using regularization. While regularization for high-dimensional problems is by now well understood, the iterative EM algorithm requires a careful balancing of making progress towards the solution while identifying the right structure (e.g., sparsity or low-rank). In particular, regularizing the M-step using the state-of-the-art high-dimensional prescriptions (e.g., `a la [19]) is not guaranteed to provide this balance. Our algorithm and analysis are linked in a way that reveals the balance between optimization and statistical errors. We specialize our general framework to sparse gaussian mixture models, high-dimensional mixed regression, and regression with missing variables, obtaining statistical guarantees for each of these examples.

Linear regression studies the problem of estimating a model parameter $\beta^* \in \R^p$, from $n$ observations $\{(y_i,x_i)\}_{i=1}^n$ from linear model $y_i = \langle \x_i,\beta^* \rangle + \epsilon_i$. We consider a significant generalization in which the relationship between $\langle x_i,\beta^* \rangle$ and $y_i$ is noisy, quantized to a single bit, potentially nonlinear, noninvertible, as well as unknown. This model is known as the single-index model in statistics, and, among other things, it represents a significant generalization of one-bit compressed sensing. We propose a novel spectral-based estimation procedure and show that we can recover $\beta^*$ in settings (i.e., classes of link function $f$) where previous algorithms fail. In general, our algorithm requires only very mild restrictions on the (unknown) functional relationship between $y_i$ and $\langle x_i,\beta^* \rangle$. We also consider the high dimensional setting where $\beta^*$ is sparse, and introduce a two-stage nonconvex framework that addresses estimation challenges in high dimensional regimes where $p \gg n$. For a broad class of link functions between $\langle x_i,\beta^* \rangle$ and $y_i$, we establish minimax lower bounds that demonstrate the optimality of our estimators in both the classical and high dimensional regimes.

We consider the problem of detecting an epidemic in a population where individual diagnoses are extremely noisy. We show that exclusively local, approximate knowledge of the contact network suffices to accurately detect the epidemic. The motivation for this problem is the plethora of examples (influenza strains in humans, or computer viruses in smartphones, etc.) where reliable diagnoses are scarce, but noisy data plentiful. In flu or phone-viruses, exceedingly few infected people/phones are professionally diagnosed (only a small fraction go to a doctor) but less reliable secondary signatures (e.g., people staying home, or greater-than-typical upload activity) are more readily available.

We propose FrogWild, a novel algorithm for fast approximation of high PageRank vertices, geared towards reducing network costs of running traditional PageRank algorithms. Our algorithm can be seen as a quantized version of power iteration that performs multiple parallel random walks over a directed graph. One important innovation is that we introduce a modification to the GraphLab framework that only partially synchronizes mirror vertices. This partial synchronization vastly reduces the network traffic generated by traditional PageRank algorithms, thus greatly reducing the per-iteration cost of PageRank. On the other hand, this partial synchronization also creates dependencies between the random walks used to estimate PageRank. Our main theoretical innovation is the analysis of the correlations introduced by this partial synchronization process and a bound establishing that our approximation is close to the true PageRank vector. We implement our algorithm in GraphLab and compare it against the default PageRank implementation. We show that our algorithm is very fast, performing each iteration in less than one second on the Twitter graph and can be up to 7x faster compared to the standard GraphLab PageRank implementation.

Cellular network performance can significantly benefit from direct device-to-device (D2D) communication, but interference from cochannel D2D communication limits the performance gain. In hybrid networks consisting of D2D and cellular links, finding the optimal interference management is challenging. In particular, we show that the problem of maximizing network throughput while guaranteeing predefined service levels to cellular users is non-convex and hence intractable. Instead, we adopt a distributed approach that is computationally extremely efficient, and requires minimal coordination, communication and cooperation among the nodes. The key algorithmic idea is a signaling mechanism that can be seen as a fictional pricing mechanism, that the base stations optimize and transmit to the D2D users, who then play a best response (i.e., selfishly) to this signal. Numerical results show that our algorithms converge quickly, have low overhead, and achieve a significant throughput gain, while maintaining the quality of cellular links at a predefined service level.

The history of infections and epidemics holds famous examples where understanding, containing, and ultimately treating an outbreak began with understanding its mode of spread. Influenza, HIV, and most computer viruses spread person to person, device to device, and through contact networks; Cholera, Cancer, and seasonal allergies, on the other hand, do not. In this paper, we study two fundamental questions of detection. First, given a snapshot view of a (perhaps vanishingly small) fraction of those infected, under what conditions is an epidemic spreading via contact (e.g., Influenza), distinguishable from a random illness operating independently of any contact network (e.g., seasonal allergies)? Second, if we do have an epidemic, under what conditions is it possible to determine which network of interactions is the main cause of the spread-the causative network-without any knowledge of the epidemic, other than the identity of a minuscule subsample of infected nodes? The core, therefore, of this paper, is to obtain an understanding of the diagnostic power of network information. We derive sufficient conditions that networks must satisfy for these problems to be identifiable, and produce efficient, highly scalable algorithms that solve these problems. We show that the identifiability condition we give is fairly mild, and in particular, is satisfied by two common graph topologies: the d-dimensional grid, and the Erdös-Renyi graphs.

The structured nature of video data motivates introducing video-aware decisions that make use of this structure for improved video transmission over wireless networks . In this paper, we introduce an architecture for real-time video transmission over multiple-input multiple-output (MIMO) wireless communication systems using loss visibility side information . We quantify the perceptual importance of a packet through the packet loss visibility and use the loss visibility distribution to provide a notion of relative packet importance . To jointly achieve high video quality and low latency, we define the optimization objective function as the throughput weighted by the loss visibility of each packet, a proxy for the total perceptual value of successful packets per unit of time. We solve the problem of mapping video packets to MIMO subchannels and adapting per-stream rates to maximize the proposed objective . We show that the solution enables jointly reaping gains in terms of improved video quality and lower latency. Optimized packet-stream mapping enables transmission of more relevant packets over more reliable streams while unequal modulation opportunistically increases the transmission rate on the stronger streams to enable low latency delivery of high priority packets. Tested on H.264-encoded video sequences, for a 4 ×4 MIMO system with three spatial streams, the proposed architecture achieves 8 dB power reduction for the same video quality and supports 2.4× higher throughput due to unequal modulation. Furthermore, the gains are achieved at the expense of few bits of cross-layer overhead rather than a complex cross-layer design.

We consider the mixed regression problem with two components, under adversarial and stochastic noise. We give a convex optimization formulation that provably recovers the true solution, and provide upper bounds on the recovery errors for both arbitrary noise and stochastic noise settings. We also give matching minimax lower bounds (up to log factors), showing that under certain assumptions, our algorithm is information-theoretically optimal. Our results represent the first (and currently only known) tractable algorithm guaranteeing successful recovery with tight bounds on recovery errors and sample complexity.

The challenge of equation-based analog synthesis comes from its dual nature: functions producing good least-square fits to SPICE-generated data are non-convex, hence not amenable to efficient optimization. In this paper, we leverage recent progress on Semidefinite Programming (SDP) relaxations of polynomial (non-convex) optimization. Using a general polynomial allows for much more accurate fitting of SPICE data compared to the more restricted functional forms. Recent SDP techniques for convex relaxations of polynomial optimizations are powerful but alone still insufficient: even for small problems, the resulting relaxations are prohibitively high dimensional.

In this paper, we develop a tractable and accurate framework for a Device-to-Device (D2D) enabled downlink cellular network with a dedicated spectrum approach, meaning that D2D links use a frequency band orthogonal to the cellular users. Using stochastic geometry, we provide accurate expressions for SINR distributions and average rates, under an assumption of interference randomization via time and/or frequency hopping. The obtained analytical results allow us to easily explore and optimize the impact of system parameters. For example, we find the optimal frequency hopping probability for the D2D users, i.e. how often they should randomly access a subband. We also propose an optimization approach for mode selection, i.e. when should potential D2D users transmit directly, and when should they fall back to the cellular mode. This can be viewed as an optimized lower bound to other more sophisticated scheduling schemes.

Given a graph, the Densest k-Subgraph (\DkS) problem asks for the subgraph on k vertices that contains the largest number of edges. In this work, we develop a novel algorithm for \DkS that searches a low-dimensional space for provably good solutions. We obtain provable performance bounds that depend on the graph spectrum. One of our results is that if there exists a k-subgraph that contains a constant fraction of all the edges, we can approximate \DkS within a factor arbitrarily close to two in polynomial time. Our algorithm runs in nearly linear time, under spectral assumptions satisfied by most graphs found in applications. Moreover, it is highly scalable and parallelizable. We demonstrate this by implementing it in MapReduce and executing numerous experiments on massive real-world graphs that have up to billions of edges. We empirically show that our algorithm can find subgraphs of significantly higher density compared to the previous state of the art.

Mixed linear regression involves the recovery of two (or more) unknown vectors from unlabeled linear measurements; that is, where each sample comes from exactly one of the vectors, but we do not know which one. It is a classic problem, and the natural and empirically most popular approach to its solution has been the EM algorithm. As in other settings, this is prone to bad local minima; however, each iteration is very fast (alternating between guessing labels, and solving with those labels). In this paper we provide a new initialization procedure for EM, based on finding the leading two eigenvectors of an appropriate matrix. We then show that with this, a re-sampled version of the EM algorithm provably converges to the correct vectors, under natural assumptions on the sampling distribution, and with nearly optimal (unimprovable) sample complexity. This provides not only the first characterization of EM’s performance, but also much lower sample complexity as compared to both standard (randomly initialized) EM, and other methods for this problem.

SRAM cell design is driven by the need to satisfy several stability and performance criteria for all cells in the array in an energy-efficient manner. Significant randomness of FET threshold voltages makes achieving this difficult and limits both the minimum cell size and minimum array supply voltage. Post-silicon adaptivity in the form of an adaptive-voltage scheme in a partitioned SRAM array can be used to reduce impact of variability despite lack of any spatial correlation in realizations. This paper develops a novel optimization flow for yield-aware cell sizing and voltage selection under variability given the availability of post-silicon voltage tuning. We formulate a two-stage stochastic optimization problem in which the first-stage decision is to select cell size and possible voltage levels, and the second-stage decision is to assign each partition to an optimal voltage after manufacturing. We develop closed-form statistical models of array margin behavior and yield as a function of Vdd, cell size, and array size. We solve the problem using dynamic programming that minimizes power while meeting yield constraints on read, write, and static noise margins. The proposed flow allows designs that are on average 8% and up to 17% more power-efficient than the designs in which voltages are selected uniformly. The results also indicate that at high-yield levels power savings can be up to 32% in the active mode and 71% in the standby mode.

Newly developed hypertext transfer protocol (HTTP)-based video streaming technologies enable flexible rate-adaptation under varying channel conditions. Accurately predicting the users' quality of experience (QoE) for rate-adaptive HTTP video streams is thus critical to achieve efficiency. An important aspect of understanding and modeling QoE is predicting the up-to-the-moment subjective quality of a video as it is played, which is difficult due to hysteresis effects and nonlinearities in human behavioral responses. This paper presents a Hammerstein-Wiener model for predicting the time-varying subjective quality (TVSQ) of rate-adaptive videos. To collect data for model parameterization and validation, a database of longer duration videos with time-varying distortions was built and the TVSQs of the videos were measured in a large-scale subjective study. The proposed method is able to reliably predict the TVSQ of rate adaptive videos. Since the Hammerstein-Wiener model has a very simple structure, the proposed method is suitable for online TVSQ prediction in HTTP-based streaming.

Real-time video demands quality-of-service (QoS) guarantees such as delay bounds for end-user satisfaction. Due to the stochastic nature of wireless fading channels, deterministic delay bounds are prohibitively difficult to guarantee. Instead, this paper proposes providing statistical delay guarantees using the concept of effective capacity. A multiuser setup is considered whereby different users have (possibly different) delay QoS constraints. The resource allocation policy that maximizes the sum video quality is derived and has a useful intuitive interpretation: The optimal operating point per user is such that the rate-distortion slope is the inverse of the supported video source rate per unit bandwidth, termed source spectral efficiency. Scheduling policies are also proposed to select a maximal user subset such that all selected users can meet their statistical delay requirement. Results show that QoS-aware scheduling and resource allocation enable supporting significantly more users under the same resource constraints.

Device-to-device (D2D) communication is emerging as a potentially attractive way to increase cellular network capacity and flexibility. However, as we describe, analyzing the performance of D2D schemes is non-trivial. In this paper, we consider a D2D overlaid cellular network model, that incorporates many of the leading models for the D2D links and access control. We show that using a Poisson point process (PPP) model for the base stations (BSs), and also a modified Matern hardcore point process (MHC) that can capture BS repulsion, we can extend tools and ideas from stochastic geometry to give compact expressions for important performance metrics, including outage probability. Extending the scope of stochastic geometry is itself an important goal, as this has proven difficult in the past. This allows a tractable approach to understand the performance of D2D overlaid cellular networks. Our simulation results subsequently demonstrate that both the PPP and the modified MHC give good approximations of true BS deployments. In particular, our simulations show the modified MHC process is more accurate than the PPP BS model, at the cost of some additional computational effort.

The rate distribution in heterogeneous networks (HetNets) greatly benefits from load balancing, by which mobile users are pushed onto lightly-loaded small cells despite the resulting loss in SINR. This offloading can be made more aggressive and robust if the macrocells leave a fraction of time/frequency resource blank, which reduces the interference to the offloaded users. We investigate the joint optimization of this technique - referred to in 3GPP as enhanced intercell interference coordination (eICIC) via almost blank subframes (ABSs) - with offloading in this paper. Although the joint cell association and blank resource (BR) problem is nominally combinatorial, by allowing users to associate with multiple base stations (BSs), the problem becomes convex, and upper bounds the performance versus a binary association. We show both theoretically and through simulation that the optimal solution of the relaxed problem still results in an association that is mostly binary. The optimal association differs significantly when the macrocell is on or off; in particular the offloading can be much more aggressive when the resource is left blank by macro BSs. Further, we observe that jointly optimizing the offloading with BR is important. The rate gain for cell edge users (the worst 3-10%) is very large - on the order of 5-10x - versus a naive association strategy without macrocell blanking.

Newly developed HTTP-based video streaming technology enables flexible rate-adaptation in varying channel conditions. The users' Quality of Experience (QoE) of rate-adaptive HTTP video streams, however, is not well understood. Therefore, designing QoE-optimized rate-adaptive video streaming algorithms remains a challenging task. An important aspect of understanding and modeling QoE is to be able to predict the up-to-the-moment subjective quality of video as it is played. We propose a dynamic system model to predict the time-varying subjective quality (TVSQ) of rate-adaptive videos that is transported over HTTP. For this purpose, we built a video database and measured TVSQ via a subjective study. A dynamic system model is developed using the database and the measured human data. We show that the proposed model can effectively predict the TVSQ of rate-adaptive videos in an online manner, which is necessary to be able to conduct QoE-optimized online rate-adaptation for HTTP-based video streaming.

Many models for sparse regression typically assume that the covariates are known completely, and without noise. Particularly in high-dimensional applications, this is often not the case. Worse yet, even estimating statistics of the noise (the noise covariance) can be a central challenge. In this paper we develop a simple variant of orthogonal matching pursuit (OMP) for precisely this setting. We show that without knowledge of the noise covariance, our algorithm recovers the support, and we provide matching lower bounds that show that our algorithm performs at the minimax optimal rate. While simple, this is the first algorithm that (provably) recovers support in a noise-distribution-oblivious manner. When knowledge of the noise-covariance is available, our algorithm matches the best-known \ell^2-recovery bounds available. We show that these too are min-max optimal. Along the way, we also obtain improved performance guarantees for OMP for the standard sparse regression problem with Gaussian noise.

We consider high dimensional sparse regression with arbitrary – possibly, severe or coordinated – errors in the covariates matrix. We are interested in understanding how many corruptions we can tolerate, while identifying the correct support. To the best of our knowledge, neither standard outlier rejection techniques, nor recently developed robust regression algorithms (that focus only on corrupted response variables), nor recent algorithms for dealing with stochastic noise or erasures, can provide guarantees on support recovery. As we show, neither can the natural brute force algorithm that takes exponential time to find the subset of data and support columns, that yields the smallest regression error. We explore the power of a simple idea: replace the essential linear algebraic calculation – the inner product – with a robust counterpart that cannot be greatly affected by a controlled number of arbitrarily corrupted points: the trimmed inner product. We consider three popular algorithms in the uncorrupted setting: Thresholding Regression, Lasso, and the Dantzig selector, and show that the counterparts obtained using the trimmed inner product are provably robust.

From Cholera, AIDS/HIV, and Malaria, to rumors and viral video, understanding the causative network behind an epidemic's spread has repeatedly proven critical for managing the spread (controlling or encouraging, as the case may be). Our current approaches to understand and predict epidemics rely on the scarce, but exact/reliable, expert diagnoses. This paper proposes a different way forward: use more readily available but also more noisy data with {\em many false negatives and false positives}, to determine the causative network of an epidemic. Specifically, we consider an epidemic that spreads according to one of two networks. At some point in time we see a small random subsample (perhaps a vanishingly small fraction) of those infected, along with an order-wise similar number of false positives. We derive sufficient conditions for this problem to be detectable, and provide an efficient algorithm that solves the hypothesis testing problem. We apply this model to two settings. In the first setting, we simply want to distinguish between random illness (a complete graph) and an epidemic (spread along a structured graph). In the second, we have a superposition of both of these, and we wish to detect which is the strongest component.

We consider streaming, one-pass principal component analysis (PCA), in the high-dimensional regime, with limited memory. Here, $p$-dimensional samples are presented sequentially, and the goal is to produce the $k$-dimensional subspace that best approximates these points. Standard algorithms require $O(p^2)$ memory; meanwhile no algorithm can do better than $O(kp)$ memory, since this is what the output itself requires. Memory (or storage) complexity is most meaningful when understood in the context of computational and sample complexity. Sample complexity for high-dimensional PCA is typically studied in the setting of the {\em spiked covariance model}, where $p$-dimensional points are generated from a population covariance equal to the identity (white noise) plus a low-dimensional perturbation (the spike) which is the signal to be recovered. It is now well-understood that the spike can be recovered when the number of samples, $n$, scales proportionally with the dimension, $p$. Yet, all algorithms that provably achieve this, have memory complexity $O(p^2)$. Meanwhile, algorithms with memory-complexity $O(kp)$ do not have provable bounds on sample complexity comparable to $p$. We present an algorithm that achieves both: it uses $O(kp)$ memory (meaning storage of any kind) and is able to compute the $k$-dimensional spike with $O(p \log p)$ sample-complexity -- the first algorithm of its kind. While our theoretical analysis focuses on the spiked covariance model, our simulations show that our algorithm is successful on much more general models for the data.

This paper considers the recovery of a low-rank matrix from an observed version that simultaneously contains both 1) erasures, most entries are not observed, and 2) errors, values at a constant fraction of (unknown) locations are arbitrarily corrupted. We provide a new unified performance guarantee on when minimizing nuclear norm plusl1norm succeeds in exact recovery. Our result allows for the simultaneous presence of random and deterministic components in both the error and erasure patterns. By specializing this one single result in different ways, we recover (up to poly-log factors) as corollaries all the existing results in exact matrix completion, and exact sparse and low-rank matrix decomposition. Our unified result also provides the first guarantees for 1) recovery when we observe a vanishing fraction of entries of a corrupted matrix, and 2) deterministic matrix completion.

Principal component analysis plays a central role in statistics, engineering, and science. Because of the prevalence of corrupted data in real-world applications, much research has focused on developing robust algorithms. Perhaps surprisingly, these algorithms are unequipped-indeed, unable-to deal with outliers in the high-dimensional setting where the number of observations is of the same magnitude as the number of variables of each observation, and the dataset contains some (arbitrarily) corrupted observations. We propose a high-dimensional robust principal component analysis algorithm that is efficient, robust to contaminated points, and easily kernelizable. In particular, our algorithm achieves maximal robustness-it has a breakdown point of 50% (the best possible), while all existing algorithms have a breakdown point of zero. Moreover, our algorithm recovers the optimal solution exactly in the case where the number of corrupted points grows sublinearly in the dimension.

For small cell technology to significantly increase the capacity of tower-based cellular networks, mobile users will need to be actively pushed onto the more lightly loaded tiers (corresponding to, e.g., pico and femtocells), even if they offer a lower instantaneous SINR than the macrocell base station (BS). Optimizing a function of the long-term rate for each user requires (in general) a massive utility maximization problem over all the SINRs and BS loads. On the other hand, an actual implementation will likely resort to a simple biasing approach where a BS in tier j is treated as having its SINR multiplied by a factor Aj≥ 1, which makes it appear more attractive than the heavily-loaded macrocell. This paper bridges the gap between these approaches through several physical relaxations of the network-wide association problem, whose solution is NP hard. We provide a low-complexity distributed algorithm that converges to a near-optimal solution with a theoretical performance guarantee, and we observe that simple per-tier biasing loses surprisingly little, if the bias values Ajare chosen carefully. Numerical results show a large (3.5x) throughput gain for cell-edge users and a 2x rate gain for median users relative to a maximizing received power association.

We investigate how a heterogeneous cellular network should self-organize by proposing a load-aware user association scheme. This is an important consideration, in order to move traffic off congested cells and onto more lightly loaded cells. Although the network-wide optimal association problem is NP hard, a closely related utility maximization problem can be made convex by applying relaxations on the association metric. We then address a low-complexity distributed algorithm that converges to a near-optimal solution with theoretical guarantee on its performance, requiring limited information and no coordination. This is directly related to range extension and small-cell biasing, which is how cell associations are likely to work in practice. Our load-aware association scheme provides theoretical guidance on the best “biasing factor” for different tiers of base stations. Numerical results show a 3.5x throughput gain for cell-edge users and a 2x gain for median users relative to the standard max-SINR association where a mobile connects to the strongest base station.

Sub-carrier grouping is a popular feedback reduction approach for orthogonal-frequency-division multiple-access (OFDMA) systems that has been adopted into fourth-generation standards such as 3GPP Long Term Evolution (LTE). Feedback reduction is motivated by the fact that the bandwidth expenditure in acquiring full information in a downlink OFDMA system scales as the product of the number of users and the number of OFDMA bands. As this is infeasible in most systems, sub-carrier grouping calls for users to report a single channel state value per predesignated group of OFDM bands. Such an approach would reduce the amount of feedback by a factor that is equal to the size of the group, albeit at a loss in throughput. In this paper, we propose a throughput-optimal joint sub-carrier grouping and data scheduling policy that makes decisions based on queue-lengths and channel states in each time slot. The feedback allocation or sub-carrier grouping policy, inspired by the current approach in LTE, operates under a total feedback budget and must periodically decide a sub-carrier grouping size for each user that obeys this resource constraint. However, as we show, the optimal allocation algorithm has complexity that, in general, scales exponentially in the number of users. Thus, we turn our attention to the important issue of computational efficiency and propose a greedy algorithm that allocates full feedback bandwidth to a constant-sized subset of users based on the network state. We evaluate the performance of this simple approach through extensive numerical experiments. We show that under asymmetric arrival rate settings, our greedy algorithm is within 10% of the optimal (throughput-wise) when consuming only 25% of the full feedback bandwidth while paying only a logarithmic (in the number of users) price in control overhead.

We consider a server serving a time-slotted queued system of multiple packet-based flows, where not more than one flow can be serviced in a single time slot. The flows have exogenous packet arrivals and time-varying service rates. At each time, the server can observe instantaneous service rates for only a subset of flows (selected from a fixed collection of observable subsets) before scheduling a flow in the subset for service. We are interested in queue-length aware scheduling to keep the queues short. The limited availability of instantaneous service rate information requires the scheduler to make a careful choice of which subset of service rates to sample. We develop scheduling algorithms that use only partial service rate information from subsets of channels, and that minimize the likelihood of queue overflow in the system. Specifically, we present a new joint subset-sampling and scheduling algorithm called Max-Exp that uses only the current queue lengths to pick a subset of flows, and subsequently schedules a flow using the Exponential rule. When the collection of observable subsets is disjoint, we show that Max-Exp achieves the best exponential decay rate, among all scheduling algorithms using partial information, of the tail of the longest queue in the system. To accomplish this, we introduce novel analytical techniques for studying the performance of scheduling algorithms using partial state information, that are of independent interest. These include new sample-path large deviations results for processes obtained by nonrandom, predictable sampling of sequences of independent and identically distributed random variables, which show that scheduling with partial state information yields a rate function significantly different from the case of full information. As a special case, Max-Exp reduces to simply serving the flow with the longest queue when the observable subsets are singleton flows, i.e., when there is effectively no a priori channel-state information; thus, our results show that this greedy scheduling policy is large-deviations optimal.

Computer (and human) networks have long had to contend with spreading viruses. Effectively controlling or curbing an outbreak requires understanding the dynamics of the spread. A virus that spreads by taking advantage of physical links or user-acquaintance links on a social network can grow explosively if it spreads beyond a critical radius. On the other hand, random infections (that do not take advantage of network structure) have very different propagation characteristics. If too many machines (or humans) are infected, network structure becomes essentially irrelevant, and the different spreading modes appear identical. When can we distinguish between mechanics of infection? Further, how can this be done efficiently? This paper studies these two questions. We provide sufficient conditions for different graph topologies, for when it is possible to distinguish between a random model of infection and a spreading epidemic model, with probability of misclassification going to zero. We further provide efficient algorithms that are guaranteed to work in different regimes.

Delivering high perceptual quality video over wireless channels is challenging due to the changing channel quality and the variations in the importance of one source packet to the next for the end-user's perceptual experience. Leveraging perceptual metrics in concert with link adaptation to maximize perceptual quality and satisfy real-time delay constraints is largely unexplored. We introduce an APP/MAC/PHY cross-layer architecture that enables optimizing perceptual quality for delay-constrained scalable video transmission. We propose an online QoS-to-QoE mapping technique to quantify the loss visibility of packets from each video layer using the ACK history and perceptual metrics. At the PHY layer, we develop a link adaptation technique that uses the QoS-to-QoE mapping to provide perceptually-optimized unequal error protection per layer according to packet loss visibility. At the APP layer, the source rate is adapted by selecting the set of temporal and quality layers to be transmitted based on the channel statistics, source rates, and playback buffer state. The proposed cross-layer optimization framework allows the channel to adapt at a faster time scale than the video codec. Furthermore, it provides a tradeoff between playback buffer occupancy and perceptual quality. We show that the proposed architecture prevents playback buffer starvation, provides immunity against short-term channel fluctuations, regulates the buffer size, and achieves a 30% increase in video capacity versus throughput-optimal link adaptation.

We consider two desired properties of learning algorithms: sparsity and algorithmic stability. Both properties are believed to lead to good generalization ability. We show that these two properties are fundamentally at odds with each other: A sparse algorithm cannot be stable and vice versa. Thus, one has to trade off sparsity and stability in designing a learning algorithm. In particular, our general result implies that ℓ(1)-regularized regression (Lasso) cannot be stable, while ℓ(2)-regularized regression is known to have strong stability properties and is therefore not sparse.

Equation-based optimization using geometric programming (GP) for automated synthesis of analog circuits has recently gained broader adoption. A major outstanding challenge is the inaccuracy resulting from fitting the complex behavior of scaled transistors to posynomial functions. In this paper, we advance a novel optimization strategy that explicitly handles the error of the model in the course of optimization. The innovation is in enabling the successive refinement of transistor models within gradually reducing ranges of operating conditions and dimensions. Refining via a brute force requires exponential complexity. The key contribution is the development of a framework that optimizes efficient convex formulations, while using SPICE as a feasibility oracle to identify solutions that are feasible with respect to the accurate behavior rather than the fitted model. Due to the poor posynomial fit, standard GP can return grossly infeasible solutions. Our approach dramatically improves feasibility. We accomplish this by introducing robust modeling of the fitting error's sample distribution information explicitly within the optimization. To address cases of highly stringent constraints, we introduce an automated method for identifying a true feasible solution through minimal relaxation of design targets. We demonstrate the effectiveness of our algorithm on two benchmarks: a two-stage CMOS operational amplifier and a voltage-controlled oscillator designed in TSMC 0.18 μm CMOS technology. Our algorithm is able to identify superior solution points producing uniformly better power and area values under a gain constraint with improvements of up to 50% in power and 10% in area for the amplifier design. Moreover, whereas standard GP methods produced solutions with constraint violations as large as 45%, our method finds feasible solutions.

A time-slotted queueing system for a wireless downlink with multiple flows and a single server is considered, with exogenous arrivals and time-varying channels. It is assumed that only one user can be serviced in a single time slot. Unlike much recent work on this problem, attention is drawn to the case where the server can obtain only partial information about the instantaneous state of the channel. In each time slot, the server is allowed to specify a single subset of flows from a collection of observable subsets, observe the current service rates for that subset, and subsequently pick a user to serve. The stability region for such a system is provided. An online scheduling algorithm is presented that uses information about marginal distributions to pick the subset and the Max-Weight rule to pick a flow within the subset, and which is provably throughput-optimal. In the case where the observable subsets are all disjoint, or where the subsets and channel statistics are symmetric, it is shown that a simple scheduling algorithm-Max-Sum-Queue-that essentially picks subsets having the largest squared-sum of queues, followed by picking a user using Max-Weight within the subset, is throughput-optimal.

Singular-value decomposition (SVD) [and principal component analysis (PCA)] is one of the most widely used techniques for dimensionality reduction: successful and efficiently computable, it is nevertheless plagued by a well-known, well-documented sensitivity to outliers. Recent work has considered the setting where each point has a few arbitrarily corrupted components. Yet, in applications of SVD or PCA, such as robust collaborative filtering or bioinformatics, malicious agents, defective genes, or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted. We present an efficient convex optimization-based algorithm that we call outlier pursuit, which under some mild assumptions on the uncorrupted points (satisfied, e.g., by the standard generative assumption in PCA problems) recovers the exact optimal low-dimensional subspace and identifies the corrupted points. Such identification of corrupted points that do not conform to the low-dimensional approximation is of paramount interest in bioinformatics, financial applications, and beyond. Our techniques involve matrix decomposition using nuclear norm minimization; however, our results, setup, and approach necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition, since we develop an approach to recover the correct column space of the uncorrupted matrix, rather than the exact matrix itself. In any problem where one seeks to recover a structure rather than the exact initial matrices, techniques developed thus far relying on certificates of optimality will fail. We present an important extension of these methods, which allows the treatment of such problems.

We consider a robust-optimization-driven system-level approach to interference management in a cellular broadband system operating in an interference-limited and highly dynamic regime. Here, base stations in neighboring cells (partially) coordinate their transmission schedules in an attempt to avoid simultaneous max-power transmission to their mutual cell edge. Limits on communication overhead and use of the backhaul require base station coordination to occur at a slower timescale than the customer arrival process. The central challenge is to properly structure coordination decisions at the slow timescale, as these subsequently restrict the actions of each base station until the next coordination period. Moreover, because coordination occurs at the slower timescale, the statistics of the arriving customers, e.g., the load, are typically only approximately known-thus, this coordination must be done with only approximate knowledge of statistics. We show that performance of existing approaches that assume exact knowledge of these statistics can degrade rapidly as the uncertainty in the arrival process increases. We show that a two-stage robust optimization framework is a natural way to model two-timescale decision problems. We provide tractable formulations for the base-station coordination problem and show that our formulation is robust to fluctuations (uncertainties) in the arriving load. This tolerance to load fluctuation also serves to reduce the need for frequent reoptimization across base stations, thus helping minimize the communication overhead required for system-level interference reduction. Our robust optimization formulations are flexible, allowing us to control the conservatism of the solution. Our simulations show that we can build in robustness without significant degradation of nominal performance.

Providing perceptual quality guarantees for video transmission over wireless channels is important for the end-user experience. The paper proposes an algorithm for perceptual quality optimization of scalable H.264 video with temporal and quality scalability. The algorithm supports adaptive unequal error protection so that different video layers are protected according to their relevance to video quality. For each video layer, a target packet error rate (PER) is selected such that a perceptual quality guarantee, measured using the multi-scale structural similarity (MS-SSIM) index, is satisfied. Given the target PER per video layer, the algorithm selects the modulation and coding scheme and the number of temporal and quality layers to transmit adaptively based on the channel state information (CSI) and source rates per video layer. Results show that the algorithm provides immunity against short term channel fluctuations, balances reliability and perceptual quality, reduces playback buffer starvation probability, and provides a convenient buffer management policy.

We consider the problem of broadcasting a large file over a wireless network (e.g., students in a campus). If each user who wants the file must download it from the carrier's WAN, dissemination time scales linearly. Two often-occurring facts suggest we can do better: (a) the demand for the file often spreads via a social network (e.g., Facebook); and (b) the devices predominantly used are GPS enabled, and equipped with a peer-to-peer (ad hoc) transmission mode. The premise of this paper is that (a) and (b) are often the case. Starting from here, we consider this coupled-network problem (demand on the social network; bandwidth on the wireless network) and taking advantage of the fact that the two networks have different topologies, we propose a file dissemination algorithm. In our scheme, users query their social network to find geographically nearby friends that have the desired file, and utilize the underlying ad hoc network to route the data via multi-hop transmissions. We show that for many popular models for social networks, the file dissemination time scales sublinearly with the number of users.

This paper considers the recovery of a low-rank matrix from an observed version that simultaneously contains both (a) erasures: most entries are not observed, and (b) errors: values at a constant fraction of (unknown) locations are arbitrarily corrupted. We provide a new unified performance guarantee on when a (natural) recently proposed method, based on convex optimization, succeeds in exact recovery. Our result allows for the simultaneous presence of random and deterministic components in both the error and erasure patterns. On the one hand, corollaries obtained by specializing this one single result in different ways recovers (upto poly-log factors) all the existing works in matrix completion, and sparse and low-rank matrix recovery. On the other hand, our results also provide the first guarantees for (a) deterministic matrix completion, and (b) recovery when we observe a vanishing fraction of entries of a corrupted matrix.

This paper investigates the impact of limited feedback on the uplink of a wireless network with multiple access points. The network adopts universal frequency re-use with each access point serving a set of users using orthogonal signalling. All wireless nodes are multiple-input-multiple-output-capable. Block Diagonalization, a popular technique for interference cancellation, is performed at each transmitter. Interference cancellation on the cell boundary is being considered in future wireless standards such as IEEE 802.16m and 3GPP Long Term Evolution Advanced. In such systems, the precoder is fed back to the mobile by its home base station. It follows that the rate achieved at the home as well as neighbouring access points is a function of the bandwidth of this feedback channel. We study the problem of minimizing the network-wide feedback budget subject to guaranteeing a minimum rate at each access point. We compute the optimal network-wide allocation using dynamic programming on junction trees with complexity that depends on the structure of the underlying interference graph. Unfortunately, when the interference graph forms a grid such as the traditional arrangement of base stations, this complexity becomes exponential in the number of users, N. We propose a greedy allocation algorithm with significantly-reduced complexity and with an approximation guarantee that, provably, scales as N. The constant of proportionality is shown to be a function of the precoder quantizer distortion, cost per feedback bit (for reliable delivery) and the size of the largest neighbourhood of the network.

Machine learning algorithms have recently attracted much interest for effective link adaptation due to their flexibility and ability to capture more environmental effects implicitly than classical adaptation algorithms. However, past applications are limited to rather simple configurations such as identifying channel condition or link adaptation in fixed or slowly varying channels. Recently, more sophisticated approaches using offline supervised learning have been proposed for link adaptation in complex configurations such as MIMO-OFDM. However, their time complexity and offline training phase hamper their real-world applicability. Approaches using online learning have shown good throughput performance, but the high memory requirement makes them inefficient or even impractical. In this paper, we propose a new effective online learning algorithm for link adaptation. Our computations show that the algorithm performs comparably to the existing online learning approaches, but ours requires minimal storage and time, which makes it more practical. Moreover it adapts to the change of channel distribution quickly.

Equation-based optimization using geometric programming (GP) for automated synthesis of analog circuits has recently gained broader adoption. A major outstanding challenge is the inaccuracy resulting from fitting the complex behavior of scaled transistors to posynomial functions. Fitting over a large region can be grossly inaccurate, and in fact, poor posynomial fit can lead to failure to find a true feasible solution. On the other hand, fitting over smaller regions and then selecting the best region, incurs exponential complexity. In this paper, we advance a novel optimization strategy that circumvents these dueling problems in the following manner: by explicitly handling the error of the model in the course of optimization, we find a potentially suboptimal, but feasible solution. This solution subsequently guides a range-refinement process of our transistor models, allowing us to reduce the range of operating conditions and dimensions, and hence obtain far more accurate GP models. The key contribution is in using the available oracle (SPICE simulations) to identify solutions that are feasible with respect to the accurate behavior rather than the fitted model. The key innovation is the explicit link between the fitting error statistics and the rate of the error uncertainty set increase, which we use in a robust optimization formulation to find feasible solutions. We demonstrate the effectiveness of our algorithm on a two benchmarks: a two-stage CMOS operational amplifier and a voltage controlled oscillator designed in TSMC 0.18μm CMOS technology. Our algorithm is able to identify superior solution points producing uniformly better power and area values under gain constraint with improvements of up to 50% in power and 10% in area for the amplifier design. We also demonstrate that when utilizing the models with the same level of modeling error, our method yields solutions that meet the constraints while the violations for the standard method were as high as 45% and larger than 15% for several constraints.

Singular Value Decomposition (and Principal Component Analysis) is one of the most widely used techniques for dimensionality reduction: successful and efficiently computable, it is nevertheless plagued by a well-known, well-documented sensitivity to outliers. Recent work has considered the setting where each point has a few arbitrarily corrupted components. Yet, in applications of SVD or PCA such as robust collaborative filtering or bioinformatics, malicious agents, defective genes, or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted. We present an efficient convex optimization-based algorithm we call Outlier Pursuit, that under some mild assumptions on the uncorrupted points (satisfied, e.g., by the standard generative assumption in PCA problems) recovers the exact optimal low-dimensional subspace, and identifies the corrupted points. Such identification of corrupted points that do not conform to the low-dimensional approximation, is of paramount interest in bioinformatics and financial applications, and beyond. Our techniques involve matrix decomposition using nuclear norm minimization, however, our results, setup, and approach, necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition, since we develop an approach to recover the correct column space of the uncorrupted matrix, rather than the exact matrix itself.

Lasso, orl1regularized least squares, has been explored extensively for its remarkable sparsity properties. In this paper it is shown that the solution to Lasso, in addition to its sparsity, has robustness properties: it is the solution to a robust optimization problem. This has two important consequences. First, robustness provides a connection of the regularizer to a physical property, namely, protection from noise. This allows a principled selection of the regularizer, and in particular, generalizations of Lasso that also yield convex optimization problems are obtained by considering different uncertainty sets. Second, robustness can itself be used as an avenue for exploring different properties of the solution. In particular, it is shown that robustness of the solution explains why the solution is sparse. The analysis as well as the specific results obtained differ from standard sparsity results, providing different geometric intuition. Furthermore, it is shown that the robust optimization formulation is related to kernel density estimation, and based on this approach, a proof that Lasso is consistent is given, using robustness directly. Finally, a theorem is proved which states that sparsity and algorithmic stability contradict each other, and hence Lasso is not stable.

SRAM cell design is driven by the need to satisfy static noise margin, write margin and read current margin (RCM) over all cells in the array in an energy-efficient manner. These constraints determine both the minimum cell size and supply voltage. RCM is set by the maximum read access time over the array. The randomness of transistor threshold voltages, and thus read times, makes maximum read time follow extreme order statistics, specifically, the Gumbel distribution which is characterized by long tails. Thus, the margin specification needs to be met at the high sigma corners in order to reach acceptable yield, resulting in oversizing and increased VDD. In this work, we demonstrate that a reduced-area bitcell design is achievable by reducing the impact of intra-array randomness through a new architecture that employs an adaptive voltage scheme in a partitioned SRAM array. The key idea is to be able to shift empirical distributions (realizations) of read time in a set of rows that form a single partition to meet the target. Because the partition is smaller than the whole array, the tail of the Gumbel distribution is significantly reduced. The adaptive voltage tuning policy is driven by the worst partition access time. For the blocks whose delay violates access time constraints, a higher voltage is selected out of the available set to gain yield, otherwise voltage is reduced for power saving. This permits smaller cell area and lower V DD at identical yield. The cost of adaptivity is in generation and routing of a small number (in our experiments, four) voltage levels and the area of one-per-partition set of PMOS switches. We demonstrate that through the voltage tuning architecture we propose, it is possible to obtain mean power consumption reduction on average by 21% iso-area. Alternatively, bitcell area can be reduced on average by 7% iso-power compared to the existing design strategy.

We consider the dimensionality-reduction problem (finding a subspace approximation of observed data) for contaminated data in the high dimensional regime, where the the number of observations is of the same magnitude as the number of variables of each observation, and the data set contains some (arbitrarily) corrupted observations. We propose a high-dimensional robust principal component analysis (HR-PCA) algorithm that is tractable, robust to contaminated points, and easily kernelizable. The resulting subspace has a bounded deviation from the desired one, and unlike ordinary PCA algorithms, achieves optimality in the limit case where the proportion of corrupted points goes to zero. In this extended abstract we provide the setup, our algorithm, and a statement of the main theorems, and defer all the details and proofs to the full paper.

In this paper, we derive bounds on the structural similarity (SSIM) index as a function of quantization rate for fixed-rate uniform quantization of image discrete cosine transform (DCT) coefficients under the high rate assumption. The space domain SSIM index is first expressed in terms of the DCT coefficients of the space domain vectors. The transform domain SSIM Index is then used to derive bounds on the average SSIM index as a function of quantization rate for Gaussian and Laplacian sources. As an illustrative example, uniform quantization of the DCT coefficients of natural images is considered. We show that the SSIM index between the reference and quantized images fall within the bounds for a large set of natural images. Further, we show using a simple example that the proposed bounds could be very useful for rate allocation problems in practical image and video coding applications.

MIMO-OFDM wireless systems require adaptive modulation and coding based on channel state information (CSI) to maximize throughput in changing wireless channels. Traditional adaptive modulation and coding attempts to predict the best rate available by estimating the packet error rate for each modulation and coding scheme (MCS) by using CSI, which has shown to be challenging. This paper considers supervised learning with the k-nearest neighbor (k-NN) algorithm as a new framework for adaptive modulation and coding. Practical k-NN operation is enabled through feature space dimensionality reduction using subcarrier ordering techniques based on postprocessing SNR. Simulation results of an IEEE 802.11n draft-compatible physical layer in flat and frequency selective wireless channels shows the k-NN with an ordered subcarrier feature space performs near ideal adaptation under packet error rate constraints.

In this paper, we present an algorithm for designing a linear equalizer that is optimal with respect to the structural similarity (SSIM) index. The optimization problem is shown to be non-convex, thereby making it non-trivial. The non-convex problem is first converted to a quasi-convex problem and then solved using a combination of first order necessary conditions and bisection search. To demonstrate the usefulness of this solution, it is applied to image denoising and image restoration examples. We show using these examples that optimizing equalizers for the SSIM index does indeed result in higher perceptual image quality compared to equalizers optimized for the ubiquitous mean squared error (MSE).

Minimum rank problems arise frequently in machine learning applications and are notoriously difficult to solve due to the non-convex nature of the rank objective. In this paper, we present the first online learning approach for the problem of rank minimization of matrices over polyhedral sets. In particular, we present two online learning algorithms for rank minimization- our first algorithm is a multiplicative update method based on a generalized experts framework, while our second algorithm is a novel application of the online convex programming framework (Zinkevich, 2003). In the latter, we flip the role of the decision maker by making the decision maker search over the constraint space instead of feasible points, as is usually the case in online convex programming. A salient feature of our online learning approach is that it allows us to give provable approximation guarantees for the rank minimization problem over polyhedral sets. We demonstrate the effectiveness of our methods on synthetic examples, and on the real-life application of low-rank kernel learning. 1.

We consider robust least-squares regression with feature-wise disturbance. We show that this formulation leads to tractable convex optimization problems, and we exhibit a particular uncertainty set for which the robust problem is equivalent to $\ell_1$ regularized regression (Lasso). This provides an interpretation of Lasso from a robust optimization perspective. We generalize this robust formulation to consider more general uncertainty sets, which all lead to tractable convex optimization problems. Therefore, we provide a new methodology for designing regression algorithms, which generalize known formulations. The advantage is that robustness to disturbance is a physical property that can be exploited: in addition to obtaining new formulations, we use it directly to show sparsity properties of Lasso, as well as to prove a general consistency result for robust regression problems, including Lasso, from a unified robustness perspective.

We propose an algorithm for designing linear equalizers that maximize the structural similarity (SSIM) index between the reference and restored signals. The SSIM index has enjoyed considerable application in the evaluation of image processing algorithms. Algorithms, however, have not been designed yet to explicitly optimize for this measure. The design of such an algorithm is nontrivial due to the nonconvex nature of the distortion measure. In this paper, we reformulate the nonconvex problem as a quasi-convex optimization problem, which admits a tractable solution. We compute the optimal solution in near closed form, with complexity of the resulting algorithm comparable to complexity of the linear minimum mean squared error (MMSE) solution, independent of the number of filter taps. To demonstrate the usefulness of the proposed algorithm, it is applied to restore images that have been blurred and corrupted with additive white gaussian noise. As a special case, we consider blur-free image denoising. In each case, its performance is compared to a locally adaptive linear MSE-optimal filter. We show that the images denoised and restored using the SSIM-optimal filter have higher SSIM index, and superior perceptual quality than those restored using the MSE-optimal adaptive linear filter. Through these results, we demonstrate that a) designing image processing algorithms, and, in particular, denoising and restoration-type algorithms, can yield significant gains over existing (in particular, linear MMSE-based) algorithms by optimizing them for perceptual distortion measures, and b) these gains may be obtained without significant increase in the computational complexity of the algorithm.

We prove that given a nearly log-concave distribution, in any partition of the space to two well separated sets, the measure of the points that do not belong to these sets is large. We apply this isoperimetric inequality to derive lower bounds on the generalization error in learning. We further consider regression problems and show that if the inputs and outputs are sampled from a nearly log-concave distribution, the measure of points for which the prediction is wrong by more than epsi0and less than epsi1is (roughly) linear in epsi1-epsi0, as long as epsi0is not too small, and epsi1not too large. We also show that when the data are sampled from a nearly log-concave distribution, the margin cannot be large in a strong probabilistic sense