Streaming Algorithms

Reconstructing dense 3D geometry from continuous video streams requires stable inference under a constant memory budget. Existing $O(1)$ frameworks primarily rely on a ``pure eviction'' paradigm, which suffers from significant information destruction due to binary token deletion and evaluation noise from localized, single-layer scoring. To address these bottlenecks, we propose StreamCacheVGGT, a training-free framework that reimagines cache management through two synergistic modules: Cross-Layer Consistency-Enhanced Scoring (CLCES) and Hybrid Cache Compression (HCC). CLCES mitigates activation noise by tracking token importance trajectories across the Transformer hierarchy, employing order-statistical analysis to identify sustained geometric salience. Leveraging these robust scores, HCC transcends simple eviction by introducing a three-tier triage strategy that merges moderately important tokens into retained anchors via nearest-neighbor assignment on the key-vector manifold. This approach preserves essential geometric context that would otherwise be lost. Extensive evaluations on five benchmarks (7-Scenes, NRGBD, ETH3D, Bonn, and KITTI) demonstrate that StreamCacheVGGT sets a new state-of-the-art, delivering superior reconstruction accuracy and long-term stability while strictly adhering to constant-cost constraints.

We study how depth, finite precision, state dimension, and chain-of-thought (CoT) affect the expressive power of multi-layer state-space models (SSMs). For the explicit-table $K$-function-composition problem, a canonical benchmark for sequential information propagation, we prove that any $L$-layer SSM solving $(L+3)$-function composition must satisfy $d^2p=Ω(N/L^3)$, where $d$ is the state dimension and $p$ is the per-scalar precision. Conversely, $K$-function composition is solved exactly by a $(K+1)$-layer generalized SSM with $d=1$ and $p=Θ(\log N)$. This gives a worst-case depth hierarchy for this formal problem family. We then distinguish post-input reasoning, in which all thought tokens are generated after the input, from input-interleaved reasoning, in which thought tokens may be inserted while the input stream is being read. Post-input reasoning does not circumvent our communication-based lower-bound pipeline, whereas input-interleaved reasoning admits bidirectional simulations with general deterministic one-pass streaming algorithms at the granularity of persistent memory. Finally, width and precision are not interchangeable under exact step-preserving simulation in the base affine-state model, but become interchangeable through the streaming-memory characterization once input-interleaved reasoning is allowed.

Modern data warehouses extend SQL with semantic operators that invoke large language models on each qualifying row, making per-row inference orders of magnitude more expensive than traditional SQL. Model cascades reduce this cost by routing most rows through a fast proxy model and delegating uncertain cases to an expensive oracle. Prior SUPG-style cascades, however, require a global proxy-score pass that is itself an LLM-inference workload and blocks output in pipelined query engines. They also target either precision or recall and cannot serve workloads that need both. We formalize the cascade routing problem for streaming semantic SQL with independent parallel workers and present two complementary algorithms within this model. SUPG-IT extends SUPG from single-pass, single-metric estimation to streaming execution by iteratively refining two thresholds as oracle labels accumulate across batches, and is the first streaming cascade with joint probabilistic guarantees on user-specified precision and recall at a chosen failure probability $δ$. GAMCAL replaces user-specified targets with a single tradeoff parameter $α$ between classification error and oracle cost, and learns a monotone Generalized Additive Model that calibrates proxy scores to true-positive probabilities and supplies pointwise uncertainty for stochastic routing. On six classification, filtering, and join benchmarks evaluated in a production semantic SQL engine, both algorithms reach $F_1 \geq 0.95$ at their best operating points. GAMCAL also leads all six datasets at a 20% delegation budget and reaches $F_1 \geq 0.95$ with up to 58% fewer oracle calls than LOTUS's SUPG cascade. SUPG-IT attains the highest best-case $F_1$, with a mean of 0.989 across the six datasets.

Approximate Nearest Neighbor (ANN) search and Approximate Kernel Density Estimation (A-KDE) are fundamental problems at the core of modern machine learning, with broad applications in data analysis, information systems, and large-scale decision making. In massive and dynamic data streams, a central challenge is to design compact sketches that preserve essential structural properties of the data while enabling efficient queries. In this work, we develop new sketching algorithms that achieve sublinear space and query time guarantees for both ANN and A-KDE for a dynamic stream of data. For ANN in the streaming model, under natural assumptions, we design a sublinear sketch that requires only $\mathcal{O}(n^{(1-η)(1+ρ)})$ memory by storing only a sublinear ($n^{-η}$) fraction of the total inputs, where $ρ$ is a parameter of the LSH family, and $0<η<1$. Our method supports sublinear query time, batch queries, and extends to the more general Turnstile model. While earlier works have focused on Exact NN, this is the first result on ANN that achieves near-optimal trade-offs between memory size and approximation error. Next, for A-KDE in the Sliding-Window model, we propose a sketch of size $\mathcal{O}\left(LW \cdot \frac{1}{\sqrt{1+ε} - 1} \log^2 N\right)$, where $L$ is the number of sketch rows, $W$ is the LSH range, $N$ is the window size, and $ε$ is the approximation error. This, to the best of our knowledge, is the first theoretical sublinear sketch guarantee for A-KDE in the Sliding-Window model. We complement our theoretical results with experiments on various real-world datasets, which show that the proposed sketches are lightweight and achieve consistently low error in practice.

Learning from a stream of experience as it arrives, also known as streaming learning, is a core part of natural learning. However, reliable streaming learning has remained a persistent challenge in modern deep reinforcement learning (RL). Instead, most deep RL algorithms learn from old experience by storing past interactions in a buffer. We show that both classical streaming RL, such as Q-learning and actor-critic, when used with deep neural networks, and batch deep RL, such as PPO, SAC, and DQN, when adapted to the streaming setting, often fail to learn. Across 58 Atari games and 50 continuous-control tasks, we find that these methods, in aggregate, perform close to random policies despite extensive task-specific hyperparameter searches. We call this pattern stream barrier. Here, we introduce Stream-X, a shared recipe for streaming deep RL algorithms that combines signal normalization, representation stabilization, and controlled parameter updates. By applying Stream-X to several base streaming RL algorithms, we provide the first family of deep RL algorithms to overcome the stream barrier. Using one prescribed hyperparameter configuration per algorithm across tasks, Stream-X substantially improves aggregate performance, often on par with batch RL algorithms. Beyond these benchmarks, we demonstrate learning with Stream-X algorithms under nonstationarity and resource constraints. Stream-AC, one of the Stream-X algorithms, repeatedly recovers performance across alternating floor-friction regimes in simulation, outperforming the evaluated PPO and SAC baselines. It also learns a heading tracking task on a robot using proprioceptive and visual features from the on-board camera in a naturally changing laboratory environment. Stream-Q learns a Pong game from pixels directly on an ESP32-S3 microcontroller, a device with limited compute and memory.

We consider streaming principal component analysis when the stochastic data generating model is subject to perturbations. While existing models assume a fixed covariance, we adopt a robust perspective where the covariance matrix belongs to a temporal uncertainty set. Under this setting, we provide fundamental limits on convergence of any algorithm recovering principal components. We analyze the convergence of the noisy power method and Oja's algorithm, both studied for the stationary data generating model, and argue that the noisy power method is rate-optimal in our setting. Finally, we demonstrate the validity of our analysis through numerical experiments on synthetic and real-world datasets.