NETWORK INSENSITIVITY TO PARAMETER NOISE VIA PARAMETER ATTACK DURING TRAINING

1 INTRODUCTION . There is increasing interest in NN and ML inference on IoT and embedded devices , which imposes energy constraints due to small battery capacity and untethered operation . Existing edge inference solutions based on CPUs or vector processing engines such as GPUs or TPUs are improving in energy efficiency , but still entail considerable energy cost ( Huang et al. , 2009 ) . Alternative compute architectures such as memristor crossbar arrays and mixed-signal event-driven neural network accelerators promise significantly reduced energy consumption for edge inference tasks . Novel non-volatile memory technologies such as resistive RAM and phase-change materials ( Chen , 2016 ; Yu & Chen , 2016 ) promise increased memory density with multiple bits per memory cell , as well as compact compute-in-memory for NN inference tasks ( Sebastian et al. , 2020 ) . Analog implementations of neurons and synapses , coupled with asynchronous digital routing fabrics , permit high sparsity in both network architecture and activity , thereby reducing energy costs associated with computation . However , both of these novel compute fabrics introduce complexity in the form of computational non-idealities , which do not exist for pure synchronous digital solutions . Some novel memory technologies support several bits per memory cell , but with uncertainty about the precise value stored on each cycle ( Le Gallo et al. , 2018b ; Wu et al. , 2019 ) . Others exhibit significant drift in stored states ( Joshi et al. , 2020 ) . Inference processors based on analog and mixed-signal devices ( Neckar et al. , 2019 ; Moradi et al. , 2018 ; Cassidy et al. , 2016 ; Schemmel et al. , 2010 ; Khaddam-Aljameh et al. , 2022 ) exhibit parameter variation across the surface of a chip , and between chips , due to manufacturing process non-idealities . Collectively these processes known as “ device mismatch ” manifest as frozen parameter noise in weights and neuron parameters . In all cases the mismatch between configured and implemented network parameters degrades the task performance by modifying the resulting mapping between input and output . Existing solutions for deploying networks to inference devices that exhibit mismatch mostly focus on per-device calibration or re-training ( Ambrogio et al. , 2018 ; Bauer et al. , 2019 ; Nandakumar et al. , 2020a ) . However , this , and other approaches such as few-shot learning or meta learning entail significant per-device handling costs , making them unfit for commercial deployment . We consider a network to be “ robust ” if the output of a network to a given input does not change in the face of parameter perturbation . With this goal , network architectures that are intrinsically robust against device mismatch can be investigated ( Thakur et al. , 2018 ; Büchel et al. , 2021 ) . Another approach is to introduce parameter perturbations during training that promote robustness during inference , for example via random pruning ( dropout ) ( Srivastava et al. , 2014 ) or by injecting noise ( Murray & Edwards , 1994 ) . In this paper we introduce a novel solution , by applying adversarial training approaches to parameter mismatch . Most existing adversarial training methods attack the input space . Here we describe an adversarial attack during training that seeks the parameter perturbation that causes the maximum degradation in network response . In summary , we make the following contributions : • We propose a novel algorithm for gradient-based supervised training of networks that are robust against parameter mismatch , by performing adversarial training in the weight space . • We demonstrate that our algorithm flattens the weight-loss landscape and therefore leads to models that are inherently more robust to parameter noise . • We show that our approach outperforms existing methods in terms of robustness . • We validate our algorithm on a highly accurate Phase Change Memory ( PCM ) -based Computein-Memory ( CiM ) simulator and achieve new state-of-the-art results in terms of performance and performance retention over time . 2 RELATED WORK . Research to date has focused mainly on adversarial attacks in the input space . With an increasing number of adversarial attacks , an increasing number of schemes defending against those attacks have been proposed ( Wang et al. , 2020 ; Zhang et al. , 2019 ; Madry et al. , 2019 ; Moosavi-Dezfooli et al. , 2018 ) . In contrast , adversarial attacks in parameter space have received little attention . Where parameter-space adversaries have been examined , it has been to enhance performance in semisupervised learning ( Cicek & Soatto , 2019 ) , to improve robustness to input-space adversarial attacks ( Wu et al. , 2020 ) , or to improve generalisation capability ( Zheng et al. , 2020 ) . We define “ robustness ” to mean that the network output should change only minimally in the face of a parameter perturbation — in other words , the weight-loss landscape should be as flat as possible at a loss minimum . Other algorithms that promote flat loss landscapes may therefore also be useful to promote robustness to parameter perturbations . Dropout ( Srivastava et al. , 2014 ) is a widely used method to reduce overfitting . During training , a random subset of units are chosen with some probability , and these units are pruned from the network for a single trial or batch . This results in the network learning to distribute its computation across many units , and acts as a regularization against overfitting . Entropy-SGD ( Chaudhari et al. , 2019 ) is a network optimisation method that minimises the local entropy around a solution in parameter space . This results in a smoothed parameter-loss landscape that should penalize sharp minima . Adversarial Block Coordinate Descent ( ABCD ) ( Cicek & Soatto , 2019 ) was proposed in order to complement input-space smoothing with weight-space smoothing in semi-supervised learning . ABCD repeatedly picks half of the network weights and performs one step of gradient ascent on them , followed by applying gradient descent on the other half . Adversarial Weight Perturbation ( AWP ) ( Wu et al. , 2020 ) was designed to improve the robustness of a network to adversarial attacks in the input space . The authors use Projected Gradient Ascent ( PGA ) on the network parameters to approximate a worst case perturbation of the weights Θ′ . PGA repeatedly computes the gradient of a loss function and updates the parameters in the direction of the ( positive ) gradient . After each update , the parameters are projected back onto a ball ( e.g . in l2 ) around the original parameters to ensure that a maximum distance is kept . Having identified an adversarial perturbation in the weight-space , an adversarial perturbation in the input-space is also found using PGA . Finally , the original weights Θ are updated using the gradient of the loss evaluated at the adversarial perturbation Θ′ . Adversarial Model Perturbation ( AMP ) ( Zheng et al. , 2020 ) improves the generalisation of conventional neural networks by optimizing a standard loss evaluated using parameters that were perturbed adversarially using PGA . Unlike our method , ( Zheng et al. , 2020 ) did not formulate the loss function as a trade-off between performance and robustness . Furthermore , the presented algorithm , unlike our method , treats the perturbation ∆Θ to the parameters Θ as a constant during backpropagation . TRadeoff-inspired Adversarial DEfense via Surrogate-loss minimization ( TRADES ) ( Zhang et al. , 2019 ) is a method for training networks that are robust against adversarial examples in the input space . The method consists of adding a boundary loss term to the loss function that measures how the network performance changes when the input is attacked . The boundary loss does not take the labels into account , so scaling it by a factor βrob allows for a principled trade-off between the robustness and the accuracy of the network . Noise injection during the forward pass ( Murray & Edwards , 1994 ) is a simple method for increasing network robustness to parameter noise . This method adds Gaussian noise to the network parameters during the forward pass and computes weight gradients with respect to the original parameters . This method regularizes the gradient magnitudes of output units with respect to the weights , thus enforcing distributed information processing and insensitivity to parameter noise . We refer to this method as “ Forward Noise ” . A recent paper proposed a method for improving the resilience to random and targeted bit errors in SRAM cells on digital Deep Neural Network ( DNN ) accelerators ( Stutz et al. , 2021 ) . By employing adversarial or random bit flips during training , the authors significantly improved the robustness to bit perturbations , enabling the accelerators to be operated below the conventional supply voltage . 3 METHODS . We use Θ to denote the set of parameters of a neural network f ( x , Θ ) that are trainable and susceptible to mismatch . The adversarial weights are denoted Θ∗ , where Θ∗t are the adversarial weights at the t-th iteration of PGA . We denote the PGA-adversary as a function A that maps parameters Θ to attacking parameters Θ∗ . We denote a mini-batch of training examples as X with y being the corresponding ground-truth labels . ∏ Epζ ( m ) denotes the projection operator on the ζ-ellipsoid in lp space . The operator denotes elementwise multiplication . The effect of component mismatch on a network parameter can be modelled using a Gaussian distribution where the standard deviation depends on the parameter magnitude ( Joshi et al. , 2020 ; Büchel et al. , 2021 ) . In this paper we restrict ourselves to mismatch-driven perturbations in the network weights . For complex Spiking Neural Networks ( SNNs ) , “ network parameters ” can refer to additional quantities such as neuronal and synaptic time constants or spiking thresholds . Our training approach described here can be equally applied to these additional parameters . We define the value of an individual parameter when deployed on a neuromorphic chip as Θmismatch ∼ N ( Θ , diag ( ζ|Θ| ) ) ( 1 ) where ζ governs the perturbation magnitude , referred to as the “ mismatch level ” . The physics underlying the neuronal- and synaptic circuits lead to a model where the amount of noise introduced into the system depends linearly on the magnitude of the parameters . If mismatch-induced perturbations had constant standard deviation independent of weight values , one could use the weight-scale invariance of neural networks as a means to achieve robustness , by simply scaling up all network weights ( see Figure S4 ) . The linear dependence of weight magnitude and mismatch noise precludes this approach . In contrast to adversarial attacks in the input space ( Carlini & Wagner , 2016 ; Moosavi-Dezfooli et al. , 2015 ; Madry et al. , 2019 ; Goodfellow et al. , 2015 ) , our method relies on adversarial attacks in parameter space . During training , we approximate the worst case perturbation of the network parameters using PGA and update the network parameters in order to mitigate these attacks . To trade-off robustness and performance , we use a surrogate loss ( Zhang et al. , 2019 ) to capture the difference in output between the normal and attacked network . Algorithm 1 illustrates the training procedure in more detail . begin Θ∗0 ←− Θ + |Θ| R ; R ∼ N ( 0 , 1 ) for t = 1 to Nsteps do g ←− ∇Θ∗t−1Lrob ( f ( Θ , X ) , f ( Θ∗t−1 , X ) ) v ←− arg max v : ‖v‖p≤1 vT g Θ∗t ← ∏ Epζattack ( Θ∗t−1 + α v ) end Θ←− Θ− η∇ΘLnat ( ( Θ , X ) , y ) + βrobLrob ( Θ , Θ∗Nsteps , X ) end Algorithm 1 : In l∞ , v corresponds to sign ( g ) and the step size α is |Θ| ζNsteps . ∏ Epζattack ( m ) denotes the projection operator on the ζattack-ellipsoid in lp space . In l∞ this corresponds to min ( max ( m , Θ − ) , Θ + ) with = ζattack |Θ| . ζattack and βrob are hyperparameters of our model . Unlike adversarial training in the input space , where adversarial inputs can be seen as a form of data augmentation , adversarial training in the parameter space poses the following challenge : Because the parameters that are attacked are the same parameters being optimized , performing gradient descent using the same loss that was used for PGA would simply revert the previous updates and no learning would occur . ABCD circumvents this problem by masking one half of the parameters in the adversarial loop and masking the other half during the gradient descent step . However , this limits the adversary in its power , and requires multiple iterations to be performed in order to update all parameters at least once . AWP approached this problem by assuming that the gradient of the loss with respect to the attacking parameters can be used in order to update the original parameters to favor minima in flatter locations in weight-space . However , it is not clear whether this assumption always holds since the gradient of the loss with respect to the attacking parameters is not necessarily the same direction that would lead to a flatter region in the weight loss-landscape . We approach this problem slightly differently : Similar to the TRADES algorithm ( Zhang et al. , 2019 ) , our algorithm optimizes a natural ( task ) loss and a separate robustness loss . Lgen ( Θ , X , y ) = Lnat ( Θ , X , y ) + βrobLrob ( Θ , A ( Θ ) , X ) Using a different loss for capturing the susceptibility of the network to adversarial attacks enables us to simultaneously optimise for performance and robustness , without PGA interfering with the gradient descent step . In our experiments , Lrob is defined as Lrob ( Θ , A ( Θ ) , X ) = KL ( f ( Θ , X ) , f ( Θ∗ , X ) ) ( 2 ) This formulation comes with a large computational overhead since it requires computing the Jacobian JΘ∗ ( Θ ) of a complex recurrent relation between Θ and Θ∗ . To make our algorithm more efficient we assume that the Jacobian is diagonal , meaning that Θ∗ = Θ + ∆Θ for some ∆Θ given by the adversary . In l∞ , the Jacobian can then be calculated efficiently using ( see suppl . material for details ) : JΘ∗ ( Θ ) = I + diag [ sign ( Θ ) ( ζattack+ ·R1 ) Nsteps ∑Nstepst=1 sign ( ∇Θ∗tLrob ( f ( Θ , X ) , f ( Θ∗t , X ) ) ) ] By making this assumption , our algorithm effectively multiplies the original training time by the number of PGA steps , similar to ( Wu et al. , 2020 ; Cicek & Soatto , 2019 ; Zheng et al. , 2020 ) . Because component mismatch is independently proportional to the magnitude of each parameter , one has to model the space in which the adversary can search for a perturbation using an axis-aligned ellipsoid in l2 and an axis-aligned box in l∞ . Using an -ball where the radius depends linearly on the individual parameter sets ( Li et al. , 2018 ; Cicek & Soatto , 2019 ; Wu et al. , 2020 ) would either give the adversary too little or too much attack space . Projecting onto an axis-aligned ellipsoid in l2 corresponds to solving the following optimization problem ( Gabay & Mercier , 1976 ; Dai , 2006 ) , which does not have a closed-form solution : x∗ = arg min x 1 2 ‖m− x‖2 s.t . ( x− c ) TW−2 ( x− c ) ≤ 1 where W = diag ( |Θ| ζ ) + I · ζconst , c = Θ and m = Θ∗ + α v. Because of the computational overhead this would incur , we only consider the l∞ case in our experiments .

Summary: This paper introduces a special loss formulation and training algorithm leveraging projected gradient ascent to build robustness to parameter noise into training for neural networks (both spiking and convolutional neural networks are tested).  The authors show that the proposed method flattens the loss landscape for small parameter perturbations with a tradeoff of increased overall cross entropy (task) loss.  It is shown that this alternative loss, when utilized during training, provides stronger resiliency to adversarially injected parameter noise than competing methods such as dropout or adversarial weight perturbation.  Tests relating to device mismatch due to manufacturing process deficiencies are also performed and the authors show their training method would be useful for building networks that don't require per-device tuning in such cases (neuromorphic/analog computing).