By taking a Bayesian perspective, we demonstrate in our latest work how biologically inspired spiking neural networks (SNNs) can exhibit learning mechanisms similar to those applied in brains, which allow them to perform continual learning. As we will see, the technique also solves a key challenge in deep learning, that is, to obtain well calibrated solutions in the face of previously unseen data.

As seen in Fig. 1, we propose to equip each synaptic weight in the SNN with a probability distribution. The distribution captures the *epistemic* uncertainty induced by the lack of knowledge of the true distribution of the data. This is done by assigning probabilities to model parameters that fit equally well the data, while also being consistent with prior knowledge. As a consequence, Bayesian learning is known to produce better calibrated decisions, i.e., decisions whose associated confidence better reflects the actual accuracy of the decision.

This contrasts with frequentist learning, in which the vector of synaptic weights is optimized by minimizing a training loss. The training loss is adopted as a proxy for the population loss, i.e., for the loss averaged over the true, unknown, distribution of the data. Therefore, frequentist learning disregards the inherent uncertainty caused by the availability of limited training data, which causes the training loss to be a potentially inaccurate estimate of the population loss. As a result, frequentist learning is known to potentially yield poorly calibrated, and overconfident decisions for ANNs.

We consider both real-valued (with possibly limited resolution, as dictated by deployment on neuromorphic hardware) and binary-valued synapses, parametrised by Gaussian and Bernoulli distributions, respectively. The advantages of models with binary-valued synapses, i.e., binary SNNs, include a reduced complexity for the computation of the membrane potential. Furthermore, binary SNNs are particularly well suited for implementations on chips with nanoscale components that provide discrete conductance levels for the synapses.

In addition to uncertainty quantification, we apply the proposed solution to continual learning, as illustrated in Fig. 2. In continual learning, the system is sequentially presented several datasets corresponding to distinct, but related, learning tasks, where each task is selected, possibly with replacement, from a pool of tasks, and its identity is unknown to the system. Its goal is to learn to make predictions that generalize well each new task, while causing minimal loss of accuracy on previous tasks.

Many existing works on continual learning draw their inspiration from the mechanisms underlying the capability of biological brains to carry out life-long learning. Learning is believed to be achieved in biological systems by modulating the strength of synaptic links. In this process, a variety of mechanisms are at work to establish short-to intermediate-term and long-term memory for the acquisition of new information over time. These mechanisms operate at different time and spatial scales.

One of the best understood mechanisms, *long-term potentiation*, contributes to the management of long-term memory through the consolidation of synaptic connections. Once established, these are rendered resistant to disruption by changing their capacity to change via *metaplasticity*. As a related mechanism, return to a base state is ensured after exposition to small, noisy changes by *heterosynaptic plasticity*, which plays a key role in ensuring the stability of neural systems. *Neuromodulation* operates at the scale of neural populations to respond to particular events registered by the brain. Finally, *episodic replay* plays a key role in the maintenance of long-term memory, by allowing biological brains to re-activate signals seen during previous active periods when inactive (i.e., sleeping).

In this work, we demonstrate how the continual learning rule we obtain exhibits some of these mechanisms. In particular, synaptic consolidation and metaplasticity for each synapse can be modeled by a *precision *parameter. A larger precision reduces the step size of the synaptic weight updates. During learning, the precision is increased to the degree that depends on the relevance of each synapse as measured by the estimated Fisher information matrix for the current mini-batch of examples.

Heterosynaptic plasticity, which drives the updates towards previously learned and resting states to prevent catastrophic forgetting, is obtained from first principles via an information risk minimization formulation with a Kullback-Leibler regularization term. This mechanism drives the updates of the precision and mean parameter towards the corresponding parameters of the variational posterior obtained at the previous task.

We start by considering the two-moons dataset shown in Fig. 3. Triangles indicate training points for a class “0’’, while circles indicate training points for a class “1”. The color intensity represents the predictive probabilities for frequentist learning and for Bayesian learning: the more intense the color, the higher the prediction confidence determined by the model. Bayesian learning is observed to provide better calibrated predictions, that are more uncertain outside the input regions covered by training data points. As can be seen, confidence for the Bayesian models can be mitigated by a parameter, as precised in the full text.

This point is further illustrated in Fig. 4 by showing the three largest probabilities assigned by the different models on selected examples from DVS-Gestures dataset, considering real-valued synapses in the top row and binary synapses in the bottom row. In the left column, we observe that, when both models predict the wrong class, Bayesian SNNs tend to do so with a lower level of certainty, and typically rank the correct class higher than their frequentist counterparts. Specifically, in the examples shown, Bayesian models with both real-valued and binary synapses rank the correct class second, while the frequentist models rank it third. Furthermore, as seen in the middle column, in a number of cases, the Bayesian models manage to predict the correct class, while the frequentist models predict a wrong class with high certainty. In the right column, we show that even when frequentist models predict the correct class and Bayesian models fail to do so, they still assign lower probabilities (i.e., <50%) to the predicted class.

Finally, we show results for continual learning on the MNIST-DVS dataset in Fig. 5. We show the evolution of the test accuracy and expected calibration error (ECE) on all tasks, represented with lines of different colors, during training. The performance on the current task is shown as a thicker line. We consider frequentist and Bayesian learning, with both real-valued and binary synapses. With Bayesian learning, the test accuracy on previous tasks does not decrease excessively when learning a new task, which shows the capacity of the technique to tackle catastrophic forgetting. Also, the ECE across all tasks is seen to remain more stable for Bayesian learning as compared to the frequentist benchmarks. For both real-valued and binary synapses, the final average accuracy and ECE across all tasks show the superiority of Bayesian over frequentist learning.

More details can be found in the full text at this link.

]]>Importantly, accuracy and calibration are two distinct criteria. As an example, Fig. 1 illustrates a QPSK demodulator trained using limited number of pilots. Depending on the input, the trained probabilistic model may result in either accurate or inaccurate demodulation decisions, whose uncertainty is either correctly or incorrectly characterized.

The property of “knowing what the AI knows/ does not know” is very useful when the AI module is used as part of a larger engineering system. In fact, well-calibrated decisions should be treated differently depending on their confidence level. Furthermore, well-calibrated models enable monitoring – by tracking the confidence of the decisions made by an AI – and other functionalities, such as anomaly detection [2].

In a recent paper from our group published on the IEEE Transaction on Signal Processing [3], we proposed a methodology to develop well-calibrated and efficient AI modules that are capable of fast adaptation. The methodology builds on Bayesian meta-learning.

To start, we summarize the main techniques under consideration.

**Conventional, frequentist, learning**ignores epistemic uncertainty – uncertainty caused by limited data – and tends to be overconfident in the presence of limited training samples.**Bayesian learning**captures epistemic uncertainty by optimizing a distribution in the model parameter space, rather than finding a single deterministic value as in frequentist learning. By obtaining decisions via ensembling, Bayesian predictors can account for the “opinions” of multiple models, hence providing more reliable decisions. Note that this approach is routinely used to quantify uncertainty in established fields like weather prediction [4].**Frequentist meta-learning**[5], also known as learning to learn, optimizes a shared training strategy across multiple tasks, so that it can easily adapt to new tasks. This is done by transferring knowledge from different learning tasks. As a communication system example, see Fig. 2 in which the demodulator adapts quickly with only few pilots for a new frame. While frequentist meta-learning is well-suited for adaptation purpose, its decisions tend to be overconfident, hence not improving monitoring in general.**Bayesian meta-learning**[6,7] integrates meta-learning with Bayesian learning in order to facilitate adaptation to new tasks for Bayesian learning.**Bayesian active meta-learning**[8] Active meta-learning can reduce the number of meta-training tasks. By considering streaming-fashion of availability of meta-training tasks, e.g., sequential supply of new frames from which we can online meta-learn the AI modules, we were able to effectively reduce the time required for satisfiable meta-learning via active meta-learning.

We first show the benefits of Bayesian meta-learning for monitoring purpose by examining the reliability of its decisions in terms of calibration. In Fig. 3, reliability diagrams for frequentist and Bayesian meta-learning are compared. For an ideal calibrated predictor, the accuracy level should match the self-reported confidence (dashed line in the plots). In can be easily checked that AI modules designed by Bayesian meta-learning (right part) are more reliable than the ones with Frequentist meta-learning (left part), validating the suitability of Bayesian meta-learning for monitoring purpose. Experimental results are obtained by considering a demodulation problem.

Fig. 4 demonstrates the impact of Bayesian active meta-learning that successfully reduces the number of required meta-training tasks. The results are obtained by considering an equalization problem.

[1] O-RAN Alliance, “O-RAN Working Group 2 AI/ML Workflow Description and Requirements,” ORAN-WG2. AIML. v01.02.02, vol. 1, 2.

[2] C. Ruah, O. Simeone, and B. Al-Hashimi, “Digital Twin-Based Multiple Access Optimization and Monitoring via Model-Driven Bayesian Learning,” *arXiv preprint arXiv:2210.05582*.

[3] K.M. Cohen, S. Park, O. Simeone and S. Shamai, “Learning to Learn to Demodulate with Uncertainty Quantification via Bayesian Meta-Learning,” *arXiv *https://arxiv.org/abs/2108.00785

[4] T. Palmer, “The Primacy of Doubt: From Climate Change to Quantum Physics, How the Science of Uncertainty Can Help Predict and Understand Our Chaotic World,” Oxford University Press, 2022.

[5] C. Finn, P. Abbeel, and S. Levine, “Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks,” in Proceedings of the 34th International Conference on Machine Learning, vol. 70. PMLR, 06–11 Aug 2017, pp. 1126–1135.

[6] J. Yoon, T. Kim, O. Dia, S. Kim, Y. Bengio, and S. Ahn, “Bayesian Model-Agnostic Meta-Learning,” Proc. Advances in neural information processing systems (NIPS), in Montreal, Canada, vol. 31, 2018.

[7] C. Nguyen, T.-T. Do, and G. Carneiro, “Uncertainty in Model-Agnostic Meta-Learning using Variational Inference,” in Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, 2020, pp. 3090–3100.

[8] J. Kaddour, S. Sæmundsson et al., “Probabilistic Active Meta-Learning,” Proc. Advances in Neural Information Processing Systems (NIPS) as Virtual-only Conference, vol. 33, pp. 20 813–20 822, 2020.

[9] C. Guo, G. Pleiss, Y. Sun, and K. Q. Weinberger, “On Calibration of Modern Neural Networks,” in International Conference on Machine Learning. PMLR, 2017, pp. 1321–1330.

]]>In our latest work, accepted for presentation at the IEEE MLSP, we are interested in training **Bayesian binary neural networks**, i.e., classical neural networks with stochastic binary weights, in a sample-efficient manner by means of meta-learning, as illustrated in Fig. 1. The key idea of this work is to model the distribution of the binary weights via a **Born machine**, i.e., via a probabilistic parametric quantum circuit (PQC), due to the capacity of PQCs to efficiently implement complex probability distributions [1]-[4]. We propose a novel method that integrates **meta-learning** with the gradient-based optimization of quantum Born machines [3], with the aim of speeding up adaptation to new learning tasks from few examples.

A Born machine produces random binary strings , where denotes the total number of model parameters, by measuring the output of a PQC defined by parameters .

As illustrated in Fig. 2, the PQC takes the initial state of *n* qubits as an input, and operates on it via a sequence of unitary gates described by a unitary matrix . This operation outputs the final quantum state

which is measured in the computational basis to produce a random binary string . Note that each basis vector of the computational basis corresponds to one of all the possible *2^n* patterns of model parameters .

The PQC can be implemented using a **hardware-efficient ansatz** [2], in which a layer of one-qubit unitary gates, parametrized by vector , is followed by a layer of fixed, entangling, two-qubit gates. This pattern can be repeated any number of times, building a progressively deeper circuit. Another option is using the **mean-field ansatz** that does not use entangling gates, and only relies on one-qubit gates.

By Born’s rule (hence the name of the circuit), the probability distribution of the output model parameter vector is given by

Importantly, Born machines only provide samples, while the actual distribution above can only be estimated by averaging multiple measurements of the PQC’s outputs. Therefore, Born machines model **implicit distributions,** and only define a stochastic procedure that directly generates samples.

Fig. 3 illustrate the results in terms of the prediction root mean squared error (RMSE) as a function of the number of meta-training iterations. By comparison with conventional per-task learning, the figure illustrates the capacity of both joint learning and meta-learning to transfer knowledge from the meta-training to the meta-test task, with hardware-efficient (HE) and mean-field (MF) quantum meta-learning clearly outperforming joint learning. For example, HE meta-learning requires around *150* meta-training iterations to achieve the same RMSE ideal per-task training, whilst joint-learning requires more than *200* to achieve comparable performance. The HE ansatz performs best, due to the use of entangling unitaries; however, the MF ansatz approaches the minimal RMSE after *230* iterations. The classical solution based on MF Bernoulli does not achieve lower RMSE than the quantum-aided meta-learning schemes, even with joint learning.

Please see the paper for a more detailed exposition, available here.

[1] Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J.C., Barends, R., Biswas, R., Boixo, S., Brandao, F.G., Buell, D.A., et al.: Quantum supremacy using a programmable superconducting processor. Nature 574(7779), 505–510 (2019)

[2] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549(7671), 242–246 (2017)

[3] Liu, J.G., Wang, L.: Differentiable learning of quantum circuit Born machines. Physical Review A 98(6), 062324 (2018)

[4] Sweke, R., Seifert, J.P., Hangleiter, D., Eisert, J.: On the quantum versus classical learnability of discrete distributions. Quantum 5, 417 (2021)

*”I’m not happy with all the analyses that go with just classical theory, because Nature isn’t classical, dammit, and if you want to make a simulation of Nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem!”*

But how can we simulate the quantum mechanical nature of Nature? This new kind of machine would become the quantum computer, and from then on, quantum computing has been on a journey with many ups and downs. Nowadays, excitement seems to be in the air again as quantum machine learning, a hybrid research discipline that combines machine learning and quantum computing, has emerged as a potential practical use of quantum hardware. Generally, quantum machine learning methods apply classical optimization routines to select parameters that define the operation of a quantum circuit. Alternative approaches, which may be more promising in the short term, involve hybrid quantum-classical models, where classical computation, e.g., for feature extraction, is combined with quantum parametric circuits [2].

In our latest work, published in the IEEE Signal Processing Letters, we focus on the hybrid classical-quantum two-layer architecture illustrated in Fig. 1.

In it, a first layer of quantum generalized linear models (QGLMs) is followed by a second classical combining layer. The input to the first, hidden, layer is obtained via amplitude encoding (see, e.g., [3]). Several implementations of QGLM neurons have been proposed in the literature using different quantum circuits. Given a binary input sample and an N-dimensional vector of binary weights, the main goal of these circuits is to produce a stochastic binary output with probabilities which are a function of the inner product

between the input state and the amplitude-encoded binary weight vector

Different solutions, along with the resulting QGLM neuron’s response functions are given in the paper. For this hybrid model, we introduced a stochastic variational optimization (SVO) approach [4] that enables the joint training of quantum and classical layers via stochastic gradient descent. The proposed SVO-based training strategy operates in a relaxed continuous space of variational classical parameters.

We show the classification accuracy, which is defined as the ratio of the number of accurate predictions over the total number of predictions made by the model, in Fig. 2 as a function of the training iterations.

The proposed SVO scheme is seen to achieve high classification accuracy for all of the considered response functions. In particular, the QGLM using the Quadratic (Q) response function yields fastest convergence and achieves the best performance. Due to the additional bias terms resulting from the swap test routine, the QGLMs relying on the Biased quadratic (BQ) and Biased centered quadratic (BCQ) response functions are slower to learn, but ultimately converge after around 3000 training iterations.

Please see the paper for a more extensive presentation, available here

Code, alongside a tutorial, are available here

[1] R. P. Feynman et al., “Simulating physics with computers,” Int. j. Theor. phys, vol. 21, no. 6/7, 1982.

[2] A. Mari, T. R. Bromley, J. Izaac, M. Schuld, and N. Killoran, “Transfer learning in hybrid classical-quantum neural networks,” Quantum, vol. 4, p. 340, 2020.

[3] M. Schuld and F. Petruccione, Machine Learning with Quantum Computers. Springer, 2021.

[4] T. Bird, J. Kunze, and D. Barber, “Stochastic variational optimization,” arXiv preprint arXiv:1809.04855, 2018.

[5] F. Tacchino, C. Macchiavello, D. Gerace, and D. Bajoni, “An artificial neuron implemented on an actual quantum processor,” npj Quantum Information, vol. 5, no. 1, pp. 1–8, 2019

**Problem Formulation**

In conventional Bayesian learning, the *model parameter* that describes the data generating distribution is assumed to be random and is endowed with a *prior distribution*. This distribution is conventionally chosen based on prior knowledge about the problem. In contrast, Bayesian meta-learning (see Fig. 1 below) aims to *automatically *infer this prior distribution by observing data from several related tasks. The statistical relationship among the tasks is accounted for via a global latent *hyperparameter *. Specifically, the model parameter for each observed task is drawn according to a shared prior distribution with shared global hyperparameter . Following the Bayesian formalism, the hyperparameter is assumed to be random and distributed according to a *hyper-prior distribution **.*

Figure 1: Bayesian meta-learning decision problem

The data from the observed related tasks, collectively called *meta-training data*, is used to reduce the expected loss incurred on a test task. The test task is modelled as generated by an independent model parameter with the same shared hyperparameter. This model parameter underlies the generation of a test task training data, used to infer the task-specific model parameter, as well as a test data sample from the test task. The Bayesian meta-learning decision problem is to predict the label corresponding to test input feature of the test task, after observing the meta-training data and the training data of the test task.

A meta-learning decision rule thus maps the meta-training data, the test task training data and test input feature to an action space. The **Bayesian meta-risk **can be defined as the minimum expected loss incurred over all meta-learning decision rules, i.e., . In the genie-aided case when the model parameter and hyper-parameter are known, the genie-aided Bayesian risk is defined as . The epistemic uncertainty, or minimum excess risk, corresponds to the difference between the Bayesian meta-risk and Genie-aided meta-risk as .

**Main Result**

Our main result shows that under the log-loss, the minimum excess meta-risk can be exactly characterized using the conditional mutual information

where H(A|B) denotes the conditional entropy of A given B and I(A;B|C) denotes the conditional mutual information between A and B given C. This in turn implies that

More importantly, we show that the epistemic uncertainty is contributed by two levels of uncertainties – model parameter level and hyperparameter level as

which scales in the order of 1/Nm+1/m, and vanishes as both the number of observed tasks and per-task data samples go to infinity. The behavior of the bounds is illustrated for the problem of meta-learning the Bayesian neural network prior for regression tasks in the figure below.

Figure 2: Performance of MEMR and derived upper bounds as a function of number of tasks and per-task data samples

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An essential property of any wireless channel is the fact that it is a shared medium, much like the air through which sound propagates is shared among the participants of a conversation. As a result, communication engineers must deal with the resulting interference, which may substantially limit the reliability and the achievable rates in a wireless communication system. A proven remedy is to adapt the transmission power to current channel conditions, which was successfully addressed by the data-driven methodology introduced in [1] in which the power control policy is parametrized by a random edge graph neural network (REGNN).

In our recent work to be presented at SPAWC 2021, we focus on the higher-level problem of facilitating adaptation of the power control policy. We consider the case where the topology of the network varies across periods of operation of the system, with each period being in turn characterized by time-varying channel conditions. In order to facilitate fast adaptation of the power control policy — in terms of data and iteration requirements — we integrate meta-learning with REGNN training.

Our meta-learning solution leverages channel state information (CSI) data from a number of previous periods to optimize an adaptation procedure that facilitates fast adaptation on a new topology to be encountered in a future period. We specifically adopt first-order meta-learning methods, namely first-order model agnostic meta-learning (FOMAML) [2] and REPTILE [3] that parametrize the adaptation procedure via its initialization within each period. While GNNs are known to be robust to changes in the topology, the proposed integration of meta-learning and REGNNs is shown to offer significant improvements in terms of sample and iteration efficiency.

The achievable sum rate with respect to the number of CSI samples used for adaptation is illustrated in Fig. 1 for a network in which the number of transmitters and receivers changes in each period. Meta-learning, via both FOMAML and REPTILE, is seen to adapt quickly to the new topology, outperforming conventional REGNN, even when allowing for fine-tuning of the later. This significant improvement can be attributed to the variability of the topologies observed across periods in the considered scenario, which makes the joint training approach in [1] ineffective. That said, when the number of samples for adaptation is sufficiently large, conventional REGNN training as in [1] outperforms meta-learning, as the initialization obtained by meta-learning induces a more substantial bias than joint training due to the mismatch in the conditions assumed for the updates on meta-training and meta-testing tasks (i.e., the different number of samples used for meta-training and adaptation).

Please see the paper for more results and a more extensive analysis, which is available here

[1] M. Eisen and A. Ribeiro, “Optimal wireless resource allocation with random edge graph neural networks,”IEEE Transactionson Signal Processing, vol. 68, pp. 2977–2991, April, 2020.

[2] C. Finn, P. Abbeel, and S. Levine, “Model-agnostic meta-learning for fast adaptation of deep networks,” inProc. InternationalConference on Machine Learning (PMLR). Sydney, 6–11 August, 2017, pp. 1126–1135.

[3] A. Nichol, J. Achiam, and J. Schulman, “On first-order meta-learning algorithms,”arXiv preprint arXiv:1803.02999, 2018.

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Most cellular deployments rely on frequency division duplex (FDD) due to its lower latency and greater coverage potential. In FDD, uplink and downlink channels use different carrier frequencies. Therefore, as illustrated in Fig. 1, with FDD, downlink channel state information (CSI) cannot be directly obtained from uplink pilots due to a lack of full reciprocity between uplink and downlink channels.

The conventional solution to this problem is to leverage downlink training and feedback from the devices. This, however, generally causes a prohibitively large downlink and uplink overhead in massive multiple-input multiple-output (MIMO) systems owing to the need to transmit a pilot sequences of length proportional to the number of antennas.

State-of-the-art recent proposals to address the inefficiencies of this conventional solutions adopt machine learning (ML) tools. The use of ML is justified by the technical challenges arising from the lack of efficient optimal model-based methods.

In our recent work to be presented at SPAWC 2021, we contribute to this line of work by introducing a new ML-based solution that improves over the state of the art by leveraging partial channel reciprocity and the tool of hypernetworks.

In this work, we propose a novel end-to-end architecture — HyperRNN — illustrated in Fig. 2. The main innovation of the approach is that simultaneously transmitted pilot symbols in the uplink, across multiple time slots, are leveraged to automatically extract long-term reciprocal channel features (see Fig. 1) via a hypernetwork that determines the weight of the downlink CSI estimation or beamforming network. Importantly, the long-term features implicitly underlie the discriminative mapping implemented by the hypernetwork between uplink pilots and downlink CSI estimation network, rather than estimated explicitly. The second main innovation is to incorporate recurrent neural networks (RNNs), in lieu of (feedforward) deep neural networks (DNNs) for both uplink and downlink processing in order to leverage the temporal correlation of the fading amplitudes.

We compare the the normalized mean square error (NMSE) performance of our proposed HyperRNN and an earlier work based on end-to-end training procedure, downlink-based DNN (DL-DNN), which encompasses downlink pilot training, distributed quantization for the uplink and downlink channel estimation. Simulations are performed over the spatial channel model (SCM) standardized in 3GPP Release 16. Fig. 3 demonstrates the NMSE of the proposed HyperRNN and of the benchmark DL-DNN for channel estimation as a function of the number of paths. Larger performance gains can be achieved when the channel has a lower number of paths. In fact, in this regime, the invariant of the long-term features of the channel defines a low-rank structure of the channel that can be leveraged by the hypernetwork.

Full paper can be found here.

]]>Conventional learning optimizes model parameters using a training algorithm, while meta-learning optimizes the hyperparameters of a training algorithm. A meta-learner has access to data from a class of tasks, and its goal is to ensure that the resulting training algorithm, also called base-learner, performs well on any new tasks from the same class. For example, the base-learner could be a stochastic gradient descent (SGD) algorithm with hyperparameters like initialization or learning rate.

The tasks observed during meta-training are conventionally assumed to belong to a task environment, which defines a distribution over the class of tasks, where each task has an associated data distribution. The statistical properties of the task environment then determine the similarity between the tasks. Intuitively, if the average “distance” between data distributions of any two tasks in the task environment is small, the meta-learner should be able to learn a suitable shared hyperparameter by observing fewer number of tasks.

In our recent work accepted to ISIT 2021, we build on the above observation and address the following questions for a fixed base-learner and meta-learner: How to measure task similarity? Given the level of similarity of the tasks in the environment, how many tasks and how much data per task should be observed to guarantee that the target average population loss for new tasks can be well approximated using the available meta-training data?

The difference between the average population loss on a new, previously unseen, meta-test task and the meta-training loss on the data gathered from the meta-training tasks is the **meta-generalization gap**, and is a measure of the generalization capability of the meta-learner. Our main contribution is a novel **information theoretic bound** on the **average absolute value of the meta-generalization gap**, that explicitly captures the impact of task relatedness, the number of tasks, and the number of data samples per task on meta-generalization.

**Results**

** **Although information-theoretic bounds on generalization performance of meta-learning have been previously studied – in both average and high probability PAC-Bayesian settings, they fail to capture the impact of task similarity in meta-generalization gap. We identify the following distinguishing components of our analysis that enable the explicit characterization of task similarity:

- Performance metric – Earlier work on meta-learning considers the
**absolute average meta-generalization gap**( ) as the performance metric, that computes the absolute value of the average of the meta-generalization gap over selection of meta-training and meta-test tasks. By “mixing up’’ the tasks via first averaging over the tasks and then taking the absolute value, the metric fails to account for the dissimilarity between the training and test tasks.- We mitigate this drawback via a new metric, namely the
**average absolute value of the meta-generalization gap**( ). This metric first computes the absolute value of meta-generalization gap for a given selection of meta-test task and meta-training tasks, and then average it over all such selections. By doing so, this metric distinguishes the contribution of each selection of meta-training and meta-test tasks, and thus capture the role of similarity between tasks, in the generalization performance of a meta-learner. Moreover, in contrast to absolute average meta-generalization gap, this new metric is non-vanishing in the asymptotic limit of large number of tasks and per-task training samples. This clearly reflects that the meta-training loss cannot provide an asymptotically accurate estimate of meta-test loss, which is evaluated on a priori unknown task.

- We mitigate this drawback via a new metric, namely the

- Measures of Task Relatedness – A task environment is said to be epsilon- related if the average “distance” between the data distributions of any two tasks in the environment is upper bounded by epsilon. We consider KL divergence based as well as the Jensen-Shannon based distance measures. While the former can be unbounded, the latter is always bounded.

Using the above defined measures of performance and task relatedness, we obtain novel information theoretic bounds on the average absolute value of the meta-generalization gap. The obtained bound demonstrates that (a) as the task dissimilarity parameter increases, more number of meta-training tasks are required to ensure meta-generalization, and that (b) there exists a non-vanishing gap, which arises due to task dissimilarity, even in the limit of large number of meta-training tasks and meta-test tasks.

We also study examples where the obtained bound can be evaluated analytically or numerically. For the example of ridge regression with meta-learned bias, we illustrate the impact of task dissimilarity parameter on the two performance metrics, and their corresponding upper bounds, in the following figure. As can be seen, while the absolute average meta-generalization gap metric appears to be largely insensitive to task dissimilarity, our metric reveals the role of task similarity, as captured by the bounds derived in the paper.

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**Problem**

Machine learning models have become a ubiquitous tool for inference in various end-user applications such as natural language processing, face recognition and mobile healthcare management. In such highly personalized cases, it is important for the model to be able to adapt to the unique features of an individual with only a minimal amount of training data. Furthermore, the devices for which these personalized inference problems are relevant are typically power and/or memory constrained which limits the size of the model that can be used for learning on the edge.

**Solution**

In our recent work, accepted for publication in DSLW 2021, we focus on a solution using spiking neural networks (SNNs) that leverages the popular model agnostic meta-learning method to train a model that performs online inference while concurrently improving its ability to adapt to new tasks (termed online-within-online meta-learning (OWOML)-SNN). This meta-training method learns a hyperparameter initialization that can be quickly adapted to new tasks from a certain family as opposed to the standard paradigm in which a large ‘universal’ model is fixed at deployment. Biologically inspired SNN models that operate on sparse binary signals, are known to provide benefits in terms of energy efficiency, especially (as in our case) where a local learning rule would allow it to be implemented on a neuromorphic edge processor such as Intel’s Loihi chip.

As show in Fig. 1, the proposed system assumes a stream of tasks for the SNN to adapt to, with streaming within-task data that it can learn from for online inference. Data from past tasks is stored in a fixed size buffer to be used for the meta-learning update. When new data for the current task arrives, multiple SNNs are instantiated using the hyperparameter initialization. One of those SNNs is used for online learning and inference while the others implement the meta-learning update to the hyperparameter initialization for use in the next iteration.

**Results**

The performance of OWOML-SNN is compared to conventional per-task training and joint training. Under conventional training, a single SNN model is randomly initialized and learns only from the data available for the current task to perform online inference. Under joint training the SNN learns a so called ‘universal’ initialization by standard training across all previous tasks and then does per-task adaptation. We consider the family of omniglot character 2-way classification tasks with rate encoding to test the within-task generalization capabilities of our meta-learner.

The results in Fig. 2 for test accuracy over training at convergence of the hyperparameter initialization and joint training initialization show that the meta-learned initialization enables the fastest online adaptation of the three, providing an improvement in accuracy over conventional training that is about 18% larger that that of joint training after training on 5 examples from each class.

Please see our paper for further details.

Bleema Rosenfeld

]]>**Problem Description **

Federated Learning (FL) refers to distributed protocols that avoid direct raw data exchange among the participating devices while training for a common learning task. This way, FL can potentially reduce the information on the local data sets that is leaked via communications. Nevertheless, the model updates shared by the devices may still reveal information about local data. For example, a malicious server could potentially infer the presence of an individual data sample from a learnt model by membership inference attack or model inversion attack.

Differential privacy (DP) quantifies information leaked about individual data points by measuring the sensitivity of the disclosed statistics to changes in the input data set at a single data point. DP can be guaranteed by introducing a level of uncertainty into the released model that is sufficient to mask the contribution of any individual data point. The most typical approach is to add random perturbations, e.g., Gaussian. This suggests that, when FL is implemented in wireless systems, the channel noise can directly act as a privacy-inducing mechanism.

**Suggested Solution **

In recent work, we have designed differentially private wireless distributed gradient descent via the direct, uncoded, transmission of gradients from devices to edge server. The channel noise is utilized as a privacy preserving mechanism and dynamic power control is separately optimized for orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA) protocols with the goal of minimizing the learning optimality gap under privacy and power constraints across a given number of communication blocks. *Our recent work to appear in IEEE Journal on Selected Areas in Communications* tackles this problem. One of our main results shows that, as long as the privacy constraint level, measured via DP, is below a threshold that decreases with the signal-to-noise ratio (SNR), uncoded transmission achieves privacy “for free”, i.e., without affecting the learning performance. As our analysis demonstrates, channel noise added in the first iterations tends to impact convergence less significantly than the noise added in later iterations, whereas the privacy level depends on a weighted sum of the inverse noise power across the iteration. These properties, captured by compact analytical expressions derived in this paper, are leveraged for adaptive power allocation, yielding significant performance gains over standard static power allocation.

**Some Results **

The performance is first evaluated by using randomly generated synthetic dataset. In the considered range of DP level, as illustrated in the figure below, NOMA with either adaptive or static power allocation (PA) achieves better performance than OMA. Furthermore, the proposed adaptive PA obtains a significant performance gain over static PA under stringent DP constraints, while the performance advantage of adaptive PA decreases as the DP constraint is relaxed. The figure also shows the threshold values of DP level beyond which the privacy “for free”.

The performance is also evaluated by MNIST data set as summarized in the last figure. With conventional static PA, the increasing communication budget is seen to largely degrade performance. This is because more communication blocks may cause an increase in privacy loss. In contrast, adaptive PA is able to properly allocate power across the communication blocks thereby achieves a lower training loss.

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