Luca Colombo

Luca Colombo, Fabrizio Pittorino, Daniele Zambon et al.

Binary Neural Networks (BNNs), which constrain both weights and activations to binary values, offer substantial reductions in computational complexity, memory footprint, and energy consumption. These advantages make them particularly well suited for deployment on resource-constrained devices. However, training BNNs via gradient-based optimization remains challenging due to the discrete nature of their variables. The dominant approach, quantization-aware training, circumvents this issue by employing surrogate gradients. Yet, this method requires maintaining latent full-precision parameters and performing the backward pass with floating-point arithmetic, thereby forfeiting the efficiency of binary operations during training. While alternative approaches based on local learning rules exist, they are unsuitable for global credit assignment and for back-propagating errors in multi-layer architectures. This paper introduces Binary Error Propagation (BEP), the first learning algorithm to establish a principled, discrete analog of the backpropagation chain rule. This mechanism enables error signals, represented as binary vectors, to be propagated backward through multiple layers of a neural network. BEP operates entirely on binary variables, with all forward and backward computations performed using only bitwise operations. Crucially, this makes BEP the first solution to enable end-to-end binary training for recurrent neural network architectures. We validate the effectiveness of BEP on both multi-layer perceptrons and recurrent neural networks, demonstrating gains of up to +6.89% and +10.57% in test accuracy, respectively. The proposed algorithm is released as an open-source repository.

Alberto Pirillo, Luca Colombo, Manuel Roveri

Quantization is a pivotal technique for managing the growing computational and memory demands of Deep Neural Networks (DNNs). By reducing the number of bits used to represent weights and activations (typically from 32-bit Floating-Point (FP) to 16-bit or 8-bit integers), quantization reduces memory footprint, energy consumption, and execution time of DNNs. However, most existing methods typically target DNN inference, while training still relies on FP operations, limiting applicability in environments where FP arithmetic is unavailable. To date, only one prior work has addressed integer-only training, and only for Multi-Layer Perceptron (MLP) architectures. This paper introduces NITRO-D, a novel framework for training deep integer-only Convolutional Neural Networks (CNNs) that operate entirely in the integer domain for both training and inference. NITRO-D enables training of integer CNNs without requiring a separate quantization scheme. Specifically, it introduces a novel architecture that integrates multiple local-loss blocks, which include the proposed NITRO-Scaling layer and NITRO-ReLU activation function. The proposed framework also features a novel learning algorithm that employs local error signals and leverages IntegerSGD, an optimizer specifically designed for integer computations. NITRO-D is implemented as an open-source Python library. Extensive evaluations on state-of-the-art image recognition datasets demonstrate its effectiveness. For integer-only MLPs, NITRO-D improves test accuracy by up to +5.96% over the state-of-the-art. It also successfully trains integer-only CNNs, reducing memory requirements and energy consumption by up to 76.14% and 32.42%, respectively, compared to the traditional FP backpropagation algorithm.

Luca Colombo, Alessandro Falcetta, Manuel Roveri

Training machine and deep learning models directly on extremely resource-constrained devices is the next challenge in the field of tiny machine learning. The related literature in this field is very limited, since most of the solutions focus only on on-device inference or model adaptation through online learning, leaving the training to be carried out on external Cloud services. An interesting technological perspective is to exploit Federated Learning (FL), which allows multiple devices to collaboratively train a shared model in a distributed way. However, the main drawback of state-of-the-art FL algorithms is that they are not suitable for running on tiny devices. For the first time in the literature, in this paper we introduce TIFeD, a Tiny Integer-based Federated learning algorithm with Direct Feedback Alignment (DFA) entirely implemented by using an integer-only arithmetic and being specifically designed to operate on devices with limited resources in terms of memory, computation and energy. Besides the traditional full-network operating modality, in which each device of the FL setting trains the entire neural network on its own local data, we propose an innovative single-layer TIFeD implementation, which enables each device to train only a portion of the neural network model and opens the door to a new way of distributing the learning procedure across multiple devices. The experimental results show the feasibility and effectiveness of the proposed solution. The proposed TIFeD algorithm, with its full-network and single-layer implementations, is made available to the scientific community as a public repository.

Alberto Pirillo, Luca Colombo

Growing concerns over data privacy underscore the need for deep learning methods capable of processing sensitive information without compromising confidentiality. Among privacy-enhancing technologies, Homomorphic Encryption (HE) stands out by providing post-quantum cryptographic security and end-to-end data protection, safeguarding data even during computation. While Deep Neural Networks (DNNs) have gained attention in HE settings, their use has largely been restricted to encrypted inference. Prior research on encrypted training has primarily focused on logistic regression or has relied on multi-party computation to enable model fine-tuning. This stems from the substantial computational overhead and algorithmic complexity involved in DNNs training under HE. In this paper, we present ReBoot, the first framework to enable fully encrypted and non-interactive training of DNNs. Built upon the CKKS scheme, ReBoot introduces a novel HE-compliant neural network architecture based on local error signals, specifically designed to minimize multiplicative depth and reduce noise accumulation. ReBoot employs a tailored packing strategy that leverages real-number arithmetic via SIMD operations, significantly lowering both computational and memory overhead. Furthermore, by integrating approximate bootstrapping, ReBoot learning algorithm supports effective training of arbitrarily deep multi-layer perceptrons, making it well-suited for machine learning as-a-service. ReBoot is evaluated on both image recognition and tabular benchmarks, achieving accuracy comparable to 32-bit floating-point plaintext training while enabling fully encrypted training. It improves test accuracy by up to +3.27% over encrypted logistic regression, and up to +6.83% over existing encrypted DNN frameworks, while reducing training latency by up to 8.83x. ReBoot is made available to the scientific community as a public repository.

Luca Colombo, Fabrizio Pittorino, Manuel Roveri

Binary Neural Networks (BNNs) significantly reduce computational complexity and memory usage in machine and deep learning by representing weights and activations with just one bit. However, most existing training algorithms for BNNs rely on quantization-aware floating-point Stochastic Gradient Descent (SGD), limiting the full exploitation of binary operations to the inference phase only. In this work, we propose, for the first time, a fully binary and gradient-free training algorithm for multi-layer BNNs, eliminating the need for back-propagated floating-point gradients. Specifically, the proposed algorithm relies on local binary error signals and binary weight updates, employing integer-valued hidden weights that serve as a synaptic metaplasticity mechanism, thereby enhancing its neurobiological plausibility. Our proposed solution enables the training of binary multi-layer perceptrons by using exclusively XNOR, Popcount, and increment/decrement operations. Experimental results on multi-class classification benchmarks show test accuracy improvements of up to +35.47% over the only existing fully binary single-layer state-of-the-art solution. Compared to full-precision SGD, our solution improves test accuracy by up to +35.30% under the same total memory demand, while also reducing computational cost by two to three orders of magnitude in terms of the total number of Boolean gates. The proposed algorithm is made available to the scientific community as a public repository.

Pierangelo Lombardo, Alessio Boiardi, Luca Colombo et al.

The growth of domain-specific applications of semantic models, boosted by the recent achievements of unsupervised embedding learning algorithms, demands domain-specific evaluation datasets. In many cases, content-based recommenders being a prime example, these models are required to rank words or texts according to their semantic relatedness to a given concept, with particular focus on top ranks. In this work, we give a threefold contribution to address these requirements: (i) we define a protocol for the construction, based on adaptive pairwise comparisons, of a relatedness-based evaluation dataset tailored on the available resources and optimized to be particularly accurate in top-rank evaluation; (ii) we define appropriate metrics, extensions of well-known ranking correlation coefficients, to evaluate a semantic model via the aforementioned dataset by taking into account the greater significance of top ranks. Finally, (iii) we define a stochastic transitivity model to simulate semantic-driven pairwise comparisons, which confirms the effectiveness of the proposed dataset construction protocol.