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Data in today’s world is increasingly siloed across a wide variety of entities with varying resource constraints. The quality of wisdom generated from a collaborative processing of such data is substantially better if the data from all these entities is shared across each other or centralized at a nodal entity. Such data sharing and centralization is often prohibited due to stringent privacy regulations, computational constraints, communication bottlenecks, trade secrets, trust issues and competition. This necessitates development of efficient methods for distributed computation while preserving privacy to generate wisdom whose quality is on par with the case of data centralization. This thesis covers methods introduced for the same in an inter-disciplinary manner to tackle several such problems using distributed and private computation.

We introduce PolyVeil, a protocol for private Boolean summation across \\(k\\) clients that encodes private bits as permutation matrices in the Birkhoff polytope. A two-layer architecture gives the server perfect simulation-based security (statistical distance zero) while a separate aggregator faces \\#P-hard likelihood inference via the permanent and mixed discriminant. Two variants (full and compressed) differ in what the aggregator observes. We develop a finite-sample \\((\\varepsilon,\\delta)\\)-DP analysis with explicit constants. In the full variant, where the aggregator sees a doubly stochastic matrix per client, the log-Lipschitz constant grows as \\(n^4 K_t\\) and a signal-to-noise analysis shows the DP guarantee is non-vacuous only when the private signal is undetectable. In the compressed variant, where the aggregator sees a single scalar, the univariate density ratio yields non-vacuous \\(\\varepsilon\\) at moderate SNR, with the optimal decoy count balancing CLT accuracy against noise concentration. This exposes a fundamental tension. \\#P-hardness requires the full matrix view (Birkhoff structure visible), while non-vacuous DP requires the scalar view (low dimensionality). Whether both hold simultaneously in one variant remains open. The protocol needs no PKI, has \\(O(k)\\) communication, and outputs exact aggregates.

Traditional collaborative learning approaches are based on sharing of model weights between clients and a server. However, there are advantages to resource efficiency through schemes based on sharing of embeddings (activations) created from the data. Several differentially private methods were developed for sharing of weights while such mechanisms do not exist so far for sharing of embeddings. We propose Ours to learn a privacy encoding network in conjunction with a small utility generation network such that the final embeddings generated from it are equipped with formal differential privacy guarantees. These privatized embeddings are then shared with a more powerful server, that learns a post-processing that results in a higher accuracy for machine learning tasks. We show that our co-design of collaborative and private learning results in requiring only one round of privatized communication and lesser compute on the client than traditional methods. The privatized embeddings that we share from the client are agnostic to the type of model (deep learning, random forests or XGBoost) used on the server in order to process these activations to complete a task.

The mean shift (MS) is a non-parametric, density-based, iterative algorithm that has prominent usage in clustering and image segmentation. A rigorous proof for its convergence in full generality remains unknown. Two significant steps in this direction were taken in the paper \\cite{Gh1}, which proved that for \\textit{sufficiently large bandwidth}, the MS algorithm with the Gaussian kernel always converges in any dimension, and also by the same author in \\cite{Gh2}, proved that MS always converges in one dimension for kernels with differentiable, strictly decreasing, convex profiles. In the more recent paper \\cite{YT}, they have proved the convergence in more generality,\\textit{ without any restriction on the bandwidth}, with the assumption that the KDE \\(f\\) has a continuous Lipschitz gradient on the closure of the convex hull of the trajectory of the iterated sequence of the mode estimate, and also satisfies the Łojasiewicz property there. The main theoretical result of this paper is a generalization of those of \\cite{Gh1}, where we show that (1) for\\textit{ sufficiently large bandwidth} convergence is guaranteed in any dimension with \\textit{any radially symmetric and strictly positive definite kernels}. The proof uses two alternate characterizations of radially symmetric positive definite smooth kernels by Schoenberg and Bernstein \\cite{Fass}, and borrows some steps from the proofs in \\cite{Gh1}. Although the authors acknowledge that the result in that paper is more restrictive than that of \\cite{YT} due to the lower bandwidth limit, it uses a different set of assumptions than \\cite{YT}, and the proof technique is different.

Hemodialysis patients who are on donor lists for kidney transplant may get misidentified, delaying their wait time. Thus, predicting their survival time is crucial for optimizing waiting lists and personalizing treatment plans. Predicting survival times for patients often requires large quantities of high quality but sensitive data. This data is siloed and since individual datasets are smaller and less diverse, locally trained survival models do not perform as well as centralized ones. Hence, we propose the use of Federated Learning in the context of predicting survival for hemodialysis patients. Federated Learning or FL can have comparatively better performances than local models while not sharing data between centers. However, despite the increased use of such technologies, the application of FL in survival and even more, dialysis patients remains sparse. This paper studies the performance of FL for data of hemodialysis patients from NephroPlus, the largest private network of dialysis centers in India.

Social choice functions help aggregate individual preferences while differentially private mechanisms provide formal privacy guarantees to release answers of queries operating on sensitive data. However, preserving differential privacy requires introducing noise to the system, and therefore may lead to undesired byproducts. Does an increase in the level of differential privacy for releasing the outputs of social choice functions increase or decrease the level of influence and welfare, and at what rate? In this paper, we mainly address this question in more precise terms in a referendum setting with two candidates when the celebrated randomized response mechanism is used. We show that there is an inversely-proportional relation between welfare and privacy, and also influence and privacy.

Alleviating catastrophic forgetting while enabling further learning is a primary challenge in continual learning (CL). Orthogonal-based training methods have gained attention for their efficiency and strong theoretical properties, and many existing approaches enforce orthogonality through gradient projection. In this paper, we revisit orthogonality and exploit the fact that small singular values correspond to directions that are nearly orthogonal to the input space of previous tasks. Building on this principle, we introduce NESS (Null-space Estimated from Small Singular values), a CL method that applies orthogonality directly in the weight space rather than through gradient manipulation. Specifically, NESS constructs an approximate null space using the smallest singular values of each layer's input representation and parameterizes task-specific updates via a compact low-rank adaptation (LoRA-style) formulation constrained to this subspace. The subspace basis is fixed to preserve the null-space constraint, and only a single trainable matrix is learned for each task. This design ensures that the resulting updates remain approximately in the null space of previous inputs while enabling adaptation to new tasks. Our theoretical analysis and experiments on three benchmark datasets demonstrate competitive performance, low forgetting, and stable accuracy across tasks, highlighting the role of small singular values in continual learning. The code is available at https://github.com/pacman-ctm/NESS.

Low-Rank Adaptation (LoRA) is a popular technique for efficient fine-tuning of foundation models. However, applying LoRA in federated learning environments, where data is distributed across multiple clients, presents unique challenges. Existing methods rely on traditional federated averaging of LoRA adapters, resulting in inexact updates. To address this, we propose Federated Exact LoRA, or FedEx-LoRA, which adds a residual error term to the pretrained frozen weight matrix. Our approach achieves exact updates with minimal computational and communication overhead, preserving LoRA's efficiency. We evaluate the method on various models across arithmetic reasoning, commonsense reasoning, natural language understanding and natural language generation tasks, showing consistent performance gains over state-of-the-art methods across multiple settings. Through extensive analysis, we quantify that the deviations in updates from the ideal solution are significant, highlighting the need for exact aggregation. Our method's simplicity, efficiency, and broad applicability position it as a promising solution for accurate and effective federated fine-tuning of foundation models. Our code is publicly available at https://github.com/RaghavSinghal10/fedex-lora.

Federated multi-task learning (FMTL) aims to simultaneously learn multiple related tasks across clients without sharing sensitive raw data. However, in the decentralized setting, existing FMTL frameworks are limited in their ability to capture complex task relationships and handle feature and sample heterogeneity across clients. To address these challenges, we introduce a novel sheaf-theoretic-based approach for FMTL. By representing client relationships using cellular sheaves, our framework can flexibly model interactions between heterogeneous client models. We formulate the sheaf-based FMTL optimization problem using sheaf Laplacian regularization and propose the Sheaf-FMTL algorithm to solve it. We show that the proposed framework provides a unified view encompassing many existing federated learning (FL) and FMTL approaches. Furthermore, we prove that our proposed algorithm, Sheaf-FMTL, achieves a sublinear convergence rate in line with state-of-the-art decentralized FMTL algorithms. Extensive experiments show that although Sheaf-FMTL introduces computational and storage overhead due to the management of interaction maps, it achieves substantial communication savings in terms of transmitted bits when compared to decentralized FMTL baselines. This trade-off makes Sheaf-FMTL especially suitable for cross-silo FL scenarios, where managing model heterogeneity and ensuring communication efficiency are essential, and where clients have adequate computational resources.

Low-Rank Adaptation (LoRA) is a popular technique for efficient fine-tuning of foundation models. However, applying LoRA in federated learning environments, where data is distributed across multiple clients, presents unique challenges. Existing methods rely on traditional federated averaging of LoRA adapters, resulting in inexact updates. To address this, we propose Federated Exact LoRA, or FedExLoRA, which adds a residual error term to the pretrained frozen weight matrix. Our approach achieves exact updates with minimal computational and communication overhead, preserving LoRA's efficiency. We evaluate the method on various models across arithmetic reasoning, commonsense reasoning, natural language understanding and natural language generation tasks, showing consistent performance gains over state-of-the-art methods across multiple settings. Through extensive analysis, we quantify that the deviations in updates from the ideal solution are significant, highlighting the need for exact aggregation. Our method's simplicity, efficiency, and broad applicability position it as a promising solution for accurate and effective federated fine-tuning of foundation models. Our code is publicly available at https://github.com/RaghavSinghal10/fedex-lora.