Large language models (LLMs) have achieved impressive success, but their high memory requirements present challenges for long-context token generation. The memory complexity of long-context LLMs is primarily due to the need to store Key-Value (KV) embeddings in their KV cache. We present BalanceKV, a KV cache compression method based on geometric sampling process stemming from Banaszczyk's vector balancing theory, which introduces dependencies informed by the geometry of keys and value tokens, and improves precision. BalanceKV offers both theoretically proven and empirically validated performance improvements over existing methods.

In a graph bisection problem, we are given a graph $G$ with two equally-sized unlabeled communities, and the goal is to recover the vertices in these communities. A popular heuristic, known as spectral clustering, is to output an estimated community assignment based on the eigenvector corresponding to the second smallest eigenvalue of the Laplacian of $G$. Spectral algorithms can be shown to provably recover the cluster structure for graphs generated from certain probabilistic models, such as the Stochastic Block Model (SBM). However, spectral clustering is known to be non-robust to model mis-specification. Techniques based on semidefinite programming have been shown to be more robust, but they incur significant computational overheads. In this work, we study the robustness of spectral algorithms against semirandom adversaries. Informally, a semirandom adversary is allowed to ``helpfully'' change the specification of the model in a way that is consistent with the ground-truth solution. Our semirandom adversaries in particular are allowed to add edges inside clusters or increase the probability that an edge appears inside a cluster. Semirandom adversaries are a useful tool to determine the extent to which an algorithm has overfit to statistical assumptions on the input. On the positive side, we identify classes of semirandom adversaries under which spectral bisection using the _unnormalized_ Laplacian is strongly consistent, i.e., it exactly recovers the planted partitioning. On the negative side, we show that in these classes spectral bisection with the _normalized_ Laplacian outputs a partitioning that makes a classification mistake on a constant fraction of the vertices. Finally, we demonstrate numerical experiments that complement our theoretical findings.

Random Fourier features is one of the most popular techniques for scaling up kernel methods, such as kernel ridge regression. However, despite impressive empirical results, the statistical properties of random Fourier features are still not well understood. In this paper we take steps toward filling this gap. Specifically, we approach random Fourier features from a spectral matrix approximation point of view, give tight bounds on the number of Fourier features required to achieve a spectral approximation, and show how spectral matrix approximation bounds imply statistical guarantees for kernel ridge regression. Qualitatively, our results are twofold: on the one hand, we show that random Fourier feature approximation can provably speed up kernel ridge regression under reasonable assumptions. At the same time, we show that the method is suboptimal, and sampling from a modified distribution in Fourier space, given by the leverage function of the kernel, yields provably better performance. We study this optimal sampling distribution for the Gaussian kernel, achieving a nearly complete characterization for the case of low-dimensional bounded datasets. Based on this characterization, we propose an efficient sampling scheme with guarantees superior to random Fourier features in this regime.

Consider a sequence of bits where we are trying to predict the next bit from the previous bits. Assume we are allowed to say `predict 0' or `predict 1', and our payoff is $+1$ if the prediction is correct and $-1$ otherwise. We will say that at each point in time the loss of an algorithm is the number of wrong predictions minus the number of right predictions so far. In this paper we are interested in algorithms that have essentially zero (expected) loss over any string at any point in time and yet have small regret with respect to always predicting $0$ or always predicting $1$. For a sequence of length $T$ our algorithm has regret $14\epsilon T $ and loss $2\sqrt{T}e^{-\epsilon^2 T} $ in expectation for all strings. We show that the tradeoff between loss and regret is optimal up to constant factors. Our techniques extend to the general setting of $N$ experts, where the related problem of trading off regret to the best expert for regret to the 'special' expert has been studied by Even-Dar et al. (COLT'07). We obtain essentially zero loss with respect to the special expert and optimal loss/regret tradeoff, improving upon the results of Even-Dar et al (COLT'07) and settling the main question left open in their paper. The strong loss bounds of the algorithm have some surprising consequences. First, we obtain a parameter free algorithm for the experts problem that has optimal regret bounds with respect to $k$-shifting optima, i.e. bounds with respect to the optimum that is allowed to change arms multiple times. Moreover, for {\em any window of size $n$} the regret of our algorithm to any expert never exceeds $O(\sqrt{n(\log N+\log T)})$, where $N$ is the number of experts and $T$ is the time horizon, while maintaining the essentially zero loss property.

We adapt a probabilistic latent variable model, namely GaP (Gamma-Poisson), to ad targeting in the contexts of sponsored search (SS) and behaviorally targeted (BT) display advertising. We also approach the important problem of ad positional bias by formulating a one-latent-dimension GaP factorization. Learning from click-through data is intrinsically large scale, even more so for ads. We scale up the algorithm to terabytes of real-world SS and BT data that contains hundreds of millions of users and hundreds of thousands of features, by leveraging the scalability characteristics of the algorithm and the inherent structure of the problem including data sparsity and locality. Specifically, we demonstrate two somewhat orthogonal philosophies of scaling algorithms to large-scale problems, through the SS and BT implementations, respectively. Finally, we report the experimental results using Yahoos vast datasets, and show that our approach substantially outperform the state-of-the-art methods in prediction accuracy. For BT in particular, the ROC area achieved by GaP is exceeding 0.95, while one prior approach using Poisson regression yielded 0.83. For computational performance, we compare a single-node sparse implementation with a parallel implementation using Hadoop MapReduce, the results are counterintuitive yet quite interesting. We therefore provide insights into the underlying principles of large-scale learning.