Prasanjit Dubey

Prasanjit Dubey, Xiaoming Huo

In federated language modeling, $K$ nodes each hold $n$ samples but cannot pool data or exchange full-precision gradients or weights. We study the minimax rate at which a conditional distribution over $V$ tokens can be estimated when each node may upload at most $B$ bits per query in a public probe set. In federated probe-logit distillation (FPLD), each node transmits a scalar-quantized logit vector on the probe set, and an aggregator distills a global parametric student. Prior work (Dubey and Huo, 2026) establishes a high-probability KL rate $O(d/(Kn) + ρ\sqrt{V \log V / m} + K^{-1} \cdot 2^{-2B/V})$ plus optimization slack, with the bandwidth term in its trace-sharpened form. Whether this bandwidth-term rate is tight, and how the upper bound generalizes to heterogeneous per-node bandwidths, are left open. We close both gaps. First, the dithered FPLD construction has a matching single-round lower bound $Ω(K^{-1} \cdot 2^{-2B/V})$ under non-degeneracy, pinning the bandwidth-axis rate at $Θ(K^{-1} \cdot 2^{-2B/V})$. $T$-round sequential refinement with nested/scaled residual quantizers achieves $O(K^{-1} \cdot 2^{-2TB/V})$; vanilla FPLD's $T$-independent bandwidth term is suboptimal for every $T > 1$. Second, we establish a heterogeneous-bandwidth upper bound for per-node budgets $B_i$, paired with a closed-form optimal allocation $B_i^* = B_{\mathrm{tot}}/K + (V/2) \log_2(w_i / \bar{w}_g)$, a log-tilted water-filling rule that is the per-node analogue of reverse water-filling for distortion-rate optimization. A plug-in adaptive variant estimates the weights from a short warm-up phase and attains $1 + O(\sqrt{\log(K/δ)/(m T_0)})$ relative suboptimality. Synthetic n-gram simulations confirm that empirical KL is bracketed by the upper and lower bounds and that the optimal allocation strictly dominates uniform and inverse-weighted baselines under heterogeneous clipping.

Etash Kumar Guha, Prasanjit Dubey, Xiaoming Huo

In this paper, we derive a novel bound on the generalization error of Magnitude-Based pruning of overparameterized neural networks. Our work builds on the bounds in Arora et al. [2018] where the error depends on one, the approximation induced by pruning, and two, the number of parameters in the pruned model, and improves upon standard norm-based generalization bounds. The pruned estimates obtained using our new Magnitude-Based compression algorithm are close to the unpruned functions with high probability, which improves the first criteria. Using Sparse Matrix Sketching, the space of the pruned matrices can be efficiently represented in the space of dense matrices of much smaller dimensions, thereby lowering the second criterion. This leads to stronger generalization bound than many state-of-the-art methods, thereby breaking new ground in the algorithm development for pruning and bounding generalization error of overparameterized models. Beyond this, we extend our results to obtain generalization bound for Iterative Pruning [Frankle and Carbin, 2018]. We empirically verify the success of this new method on ReLU-activated Feed Forward Networks on the MNIST and CIFAR10 datasets.