Towards Tight Bounds for Streaming Attention

The attention mechanism is a cornerstone of modern transformer architectures. However, its expressive power comes at the cost of quadratic runtime and linear space usage. In particular, the classical transformer architecture explicitly stores all previously seen input elements (tokens) in order to generate the next one. The problem of implementing a transformer in limited space, known as KV cache compression, has received much interest over the past few years, spurring the development of powerful heuristics. Recent works of Haris et al, COLT'25 and Kochetkova et al, NeurIPS'25, formalized KV cache compression as the streaming attention approximation problem, providing both upper bounds (based on discrepancy theory) and information theoretic lower bounds. However, those papers left open a significant gap between the upper and lower bounds. For example, the space usage of their algorithms increases with the precision parameter, but the lower bound does not get stronger. In this work, we revisit the streaming attention approximation problem and provide nearly tight bounds on its space complexity. On the algorithmic side, we achieve the result through a surprisingly tight interplay between three distinct methods for kernel density estimation: discrepancy-based coreset constructions (e.g., Charikar-Kapralov-Waingarten'24), the polynomial method (e.g., Greengard-Rokhlin'87, Alman-Song'23), and space partitioning (e.g., Andoni-Laarhoven-Razenshteyn-Waingarten'17, Charikar-Kapralov-Nouri-Siminelakis'20). On the lower bound side, our main technical contribution is a new technique for using the INDEX problem with a large amount of side information that we hope will prove useful in other high dimensional geometric estimation problems.

Provable Quantization with Randomized Hadamard Transform

Vector quantization via random projection followed by scalar quantization is a fundamental primitive in machine learning, with applications ranging from similarity search to federated learning and KV cache compression. While dense random rotations yield clean theoretical guarantees, they require $Θ(d^2)$ time. The randomized Hadamard transform $HD$ reduces this cost to $O(d \log d)$, but its discrete structure complicates analysis and leads to weaker or purely empirical compression guarantees. In this work, we study a variant of this approach: dithered quantization with a single randomized Hadamard transform. Specifically, the quantizer applies $HD$ to the input vector and subtracts a random scalar offset before quantizing, injecting additional randomness at negligible cost. We prove that this approach is unbiased and provides mean squared error bounds that asymptotically match those achievable with truly random rotation matrices. In particular, we prove that a dithered version of TurboQuant achieves mean squared error $\bigl(π\sqrt{3}/2 + o(1)\bigr) \cdot 4^{-b}$ at $b$ bits per coordinate, where the $o(1)$ term vanishes uniformly over all unit vectors and all dimensions as the number of quantization levels grows.

Gradient-Free Approaches is a Key to an Efficient Interaction with Markovian Stochasticity

This paper deals with stochastic optimization problems involving Markovian noise with a zero-order oracle. We present and analyze a novel derivative-free method for solving such problems in strongly convex smooth and non-smooth settings with both one-point and two-point feedback oracles. Using a randomized batching scheme, we show that when mixing time $τ$ of the underlying noise sequence is less than the dimension of the problem $d$, the convergence estimates of our method do not depend on $τ$. This observation provides an efficient way to interact with Markovian stochasticity: instead of invoking the expensive first-order oracle, one should use the zero-order oracle. Finally, we complement our upper bounds with the corresponding lower bounds. This confirms the optimality of our results.

Correcting sampling biases via importance reweighting for spatial modeling

In machine learning models, the estimation of errors is often complex due to distribution bias, particularly in spatial data such as those found in environmental studies. We introduce an approach based on the ideas of importance sampling to obtain an unbiased estimate of the target error. By taking into account difference between desirable error and available data, our method reweights errors at each sample point and neutralizes the shift. Importance sampling technique and kernel density estimation were used for reweighteing. We validate the effectiveness of our approach using artificial data that resemble real-world spatial datasets. Our findings demonstrate advantages of the proposed approach for the estimation of the target error, offering a solution to a distribution shift problem. Overall error of predictions dropped from 7% to just 2% and it gets smaller for larger samples.