流式推理

When rows of an $n \times d$ matrix $A$ are given in a stream, we study algorithms for approximating the top eigenvector of $A^T A$ (equivalently, the top right singular vector of $A$). We consider worst case inputs $A$ but assume that the rows are presented to the streaming algorithm in a uniformly random order. We show that when the gap parameter $R = \sigma_1(A)^2/\sigma_2(A)^2 = \Omega(1)$, then there is a randomized algorithm that uses $O(h \cdot d \cdot \text{polylog}(d))$ bits of space and outputs a unit vector $v$ that has a correlation $1 - O(1/\sqrt{R})$ with the top eigenvector $v_1$. Here $h$ denotes the number of ``heavy rows'' in the matrix, defined as the rows with Euclidean norm at least $\|{A}\|_F/\sqrt{d \cdot \text{polylog}(d)}$. We also provide a lower bound showing that any algorithm using $O(hd/R)$ bits of space can obtain at most $1 - \Omega(1/R^2)$ correlation with the top eigenvector. Thus, parameterizing the space complexity in terms of the number of heavy rows is necessary for high accuracy solutions. Our results improve upon the $R = \Omega(\log n \cdot \log d)$ requirement in a recent work of Price. We note that Price's algorithm works for arbitrary order streams whereas our algorithm requires a stronger assumption that the rows are presented in a uniformly random order. We additionally show that the gap requirements in Price's analysis can be brought down to $R = \Omega(\log^2 d)$ for arbitrary order streams and $R = \Omega(\log d)$ for random order streams. The requirement of $R = \Omega(\log d)$ for random order streams is nearly tight for Price's analysis as we obtain a simple instance with $R = \Omega(\log d/\log\log d)$ for which Price's algorithm, with any fixed learning rate, cannot output a vector approximating the top eigenvector $v_1$.

Oja's algorithm for Streaming Principal Component Analysis (PCA) for $n$ data-points in a $d$ dimensional space achieves the same sin-squared error $O(r_{\mathsf{eff}}/n)$ as the offline algorithm in $O(d)$ space and $O(nd)$ time and a single pass through the datapoints. Here $r_{\mathsf{eff}}$ is the effective rank (ratio of the trace and the principal eigenvalue of the population covariance matrix $\Sigma$). Under this computational budget, we consider the problem of sparse PCA, where the principal eigenvector of $\Sigma$ is $s$-sparse, and $r_{\mathsf{eff}}$ can be large. In this setting, to our knowledge, *there are no known single-pass algorithms* that achieve the minimax error bound in $O(d)$ space and $O(nd)$ time without either requiring strong initialization conditions or assuming further structure (e.g., spiked) of the covariance matrix. We show that a simple single-pass procedure that thresholds the output of Oja's algorithm (the Oja vector) can achieve the minimax error bound under some regularity conditions in $O(d)$ space and $O(nd)$ time. We present a nontrivial and novel analysis of the entries of the unnormalized Oja vector, which involves the projection of a product of independent random matrices on a random initial vector. This is completely different from previous analyses of Oja's algorithm and matrix products, which have been done when the $r_{\mathsf{eff}}$ is bounded.

Bayes' rule naturally allows for inference refinement in a streaming fashion, without the need to recompute posteriors from scratch whenever new data arrives. In principle, Bayesian streaming is straightforward: we update our prior with the available data and use the resulting posterior as a prior when processing the next data chunk. In practice, however, this recipe entails i) approximating an intractable posterior at each time step; and ii) encapsulating results appropriately to allow for posterior propagation. For continuous state spaces, variational inference (VI) is particularly convenient due to its scalability and the tractability of variational posteriors, For discrete state spaces, however, state-of-the-art VI results in analytically intractable approximations that are ill-suited for streaming settings. To enable streaming Bayesian inference over discrete parameter spaces, we propose streaming Bayes GFlowNets (abbreviated as SB-GFlowNets) by leveraging the recently proposed GFlowNets --- a powerful class of amortized samplers for discrete compositional objects. Notably, SB-GFlowNet approximates the initial posterior using a standard GFlowNet and subsequently updates it using a tailored procedure that requires only the newly observed data. Our case studies in linear preference learning and phylogenetic inference showcase the effectiveness of SB-GFlowNets in sampling from an unnormalized posterior in a streaming setting. As expected, we also observe that SB-GFlowNets is significantly faster than repeatedly training a GFlowNet from scratch to sample from the full posterior.

Recent works have shown that large language model (LLM) agents are able to improve themselves from experience, which is an important ability for continuous enhancement post-deployment. However, existing benchmarks primarily evaluate their innate capabilities and do not assess their ability to improve over time. To address this gap, we introduce StreamBench, a pioneering benchmark designed to evaluate the continuous improvement of LLM agents over an input-feedback sequence. StreamBench simulates an online learning environment where LLMs receive a continuous flow of feedback stream and iteratively enhance their performance. In addition, we propose several simple yet effective baselines for improving LLMs on StreamBench, and provide a comprehensive analysis to identify critical components that contribute to successful streaming strategies. Our work serves as a stepping stone towards developing effective online learning strategies for LLMs, paving the way for more adaptive AI systems in streaming scenarios.

$\newcommand{\Tr}{\mathsf{Tr}}$ We consider the problem of high-dimensional heavy-tailed statistical estimation in the streaming setting, which is much harder than the traditional batch setting due to memory constraints. We cast this problem as stochastic convex optimization with heavy tailed stochastic gradients, and prove that the widely used Clipped-SGD algorithm attains near-optimal sub-Gaussian statistical rates whenever the second moment of the stochastic gradient noise is finite. More precisely, with $T$ samples, we show that Clipped-SGD, for smooth and strongly convex objectives, achieves an error of $\sqrt{\frac{\Tr(\Sigma)+\sqrt{\Tr(\Sigma)\\|\Sigma\\|_2}\ln(\tfrac{\ln(T)}{\delta})}{T}}$ with probability $1-\delta$, where $\Sigma$ is the covariance of the clipped gradient. Note that the fluctuations (depending on $\tfrac{1}{\delta}$) are of lower order than the term $\Tr(\Sigma)$. This improves upon the current best rate of $\sqrt{\frac{\Tr(\Sigma)\ln(\tfrac{1}{\delta})}{T}}$ for Clipped-SGD, known \emph{only} for smooth and strongly convex objectives. Our results also extend to smooth convex and lipschitz convex objectives. Key to our result is a novel iterative refinement strategy for martingale concentration, improving upon the PAC-Bayes approach of \citet{catoni2018dimension}.

Online free-viewpoint video (FVV) streaming is a challenging problem, which is relatively under-explored. It requires incremental on-the-fly updates to a volumetric representation, fast training and rendering to satisfy realtime constraints and a small memory footprint for efficient transmission. If acheived, it can enhance user experience by enabling novel applications, e.g., 3D video conferencing and live volumetric video broadcast, among others. In this work, we propose a novel framework for QUantized and Efficient ENcoding (QUEEN) for streaming FVV using 3D Gaussian Splatting (3D-GS). QUEEN directly learns Gaussian attribute residuals between consecutive frames at each time-step without imposing any structural constraints on them, allowing for high quality reconstruction and generalizability. To efficiently store the residuals, we further propose a quantization-sparsity framework, which contains a learned latent-decoder for effectively quantizing attribute residuals other than Gaussian positions and a learned gating module to sparsify position residuals. We propose to use the Gaussian viewspace gradient difference vector as a signal to separate the static and dynamic content of the scene. It acts as a guide for effective sparsity learning and speeds up training. On diverse FVV benchmarks, QUEEN outperforms the state-of-the-art online FVV methods on all metrics. Notably, for several highly dynamic scenes, it reduces the model size to just 0.7 MB per frame while training in under 5 sec and rendering at ~350 FPS.

We develop and analyze algorithms for instrumental variable regression by viewing the problem as a conditional stochastic optimization problem. In the context of least-squares instrumental variable regression, our algorithms neither require matrix inversions nor mini-batches thereby providing a fully online approach for performing instrumental variable regression with streaming data. When the true model is linear, we derive rates of convergence in expectation, that are of order $\mathcal{O}(\log T/T)$ and $\mathcal{O}(1/T^{1-\epsilon})$ for any $\epsilon>0$, respectively under the availability of two-sample and one-sample oracles respectively. Importantly, under the availability of the two-sample oracle, the aforementioned rate is actually agnostic to the relationship between confounder and the instrumental variable demonstrating the flexibility of the proposed approach in alleviating the need for explicit model assumptions required in recent works based on reformulating the problem as min-max optimization problems. Experimental validation is provided to demonstrate the advantages of the proposed algorithms over classical approaches like the 2SLS method.