SALR: Sharpness-aware Learning Rates for Improved Generalization

1 INTRODUCTION Generalization in deep learning has recently been an active area of research . The efforts to improve generalization over the past two decades have brought upon many cornerstone advances and techniques ; be it dropout ( Gal & Ghahramani , 2016 ) , batch-normalization ( Ioffe & Szegedy , 2015 ) , data-augmentation ( Shorten & Khoshgoftaar , 2019 ) , weight decay ( Loshchilov & Hutter , 2019 ) , adaptive gradient-based optimization ( Kingma & Ba , 2015 ) , architecture design and search ( Radosavovic et al. , 2020 ) , ensembles and their Bayesian counterparts ( Garipov et al. , 2018 ; Iz- mailov et al. , 2018 ) , amongst many others . Yet , recently , researchers have discovered that the concept of sharpness/flatness plays a fundamental role in generalization . Though sharpness was first discussed in the context of neural networks in the early work of Hochreiter & Schmidhuber ( 1997 ) , it was only brought to the forefront of deep learning research after the seminal paper by Keskar et al . ( 2017 ) . While trying to investigate decreased generalization performance when large batch sizes are used ( LeCun et al. , 2012 ) in stochastic gradient descent ( SGD ) , Keskar et al . ( 2017 ) notice that this phenomena can be justified by the ability of smaller batches to reach flat minimizers . Such flat minimizers in turn , generalize well as they are robust to low precision arithmetic or noise in the parameter space ( Dinh et al. , 2017 ; Kleinberg et al. , 2018 ) , as shown in Figure 1 . Since then , the generalization ability of flat minimizers has been repeatedly observed in many recent works ( Neyshabur et al. , 2017a ; Goyal et al. , 2017 ; Li et al. , 2018 ; Izmailov et al. , 2018 ) . Indeed , flat minimizers can potentially tie together many of the aforementioned approaches aimed at generalization . For instance , ( 1 ) higher gradient variance , when batches are small , increases the probability to avoid sharp regions ( same can be said for SGD compared to GD ) ( Kleinberg et al. , 2018 ) ( 2 ) averaging over multiple hypotheses leads to wider optima in ensembles and Bayesian deep learning ( Izmailov et al. , 2018 ) ( 3 ) regularization techniques such as dropout or over-parameterization can adjust the loss landscape into one that allows first order methods to favor wide valleys ( Chaudhari et al. , 2019 ; Allen-Zhu et al. , 2019 ) . In this paper we study the direct problem of developing an algorithm that can converge to flat minimizers.Specifically , we introduce SALR : a sharpness aware learning rate designed to explore the loss-surface of an objective function and avoid undesired sharp local minima . SALR dynamically updates the learning rate based on the sharpness of the neighborhood of the current solution . The idea is simple : automatically increase the learning rates at relatively sharp valleys in an effort to escape them . One of the key features of SALR is that it can be fitted into any gradient based method such as Adagrad ( Duchi et al. , 2011 ) , ADAM ( Kingma & Ba , 2015 ) and also into recent approaches towards escaping sharp valleys such as Entropy-SGD ( Chaudhari et al. , 2019 ) . 1.1 RELATED WORK . From a theoretical perspective , generalization of deep learning solutions has been explained through multiple lenses . One of which is uniform stability ( Bottou & Le Cun , 2005 ; Bottou & Bousquet , 2008 ; Hardt et al. , 2016 ; Gonen & Shalev-Shwartz , 2017 ; Bottou et al. , 2018 ) . An algorithm is uniformly stable if for all data sets differing in only one element , nearly the same outputs will be produced ( Bousquet & Elisseeff , 2002 ) . Hardt et al . ( 2016 ) show that SGD satisfies this property and derive a generalization bound for models learned with SGD . From a different viewpoint , Choromanska et al . ( 2015 ) ; Kawaguchi ( 2016 ) ; Poggio et al . ( 2017 ) ; Mohri et al . ( 2018 ) attribute generalization to the complexity of the hypothesis-space . Using measures like Rademacher complexity ( Mohri & Rostamizadeh , 2009 ) and the Vapnik-Chervonenkis ( VC ) dimension ( Sontag , 1998 ) , the former works show that deep hypothesis spaces are typically more advantageous in representing complex functions . Besides that , the importance of flatness on generalization has been theoretically highlighted through PAC-Bayes bounds ( Dziugaite & Roy , 2017 ; Neyshabur et al. , 2017b ; Wang et al. , 2018 ) . These papers highlight the ability to derive non-vacuous generalization bounds based on the sharpness of a model class while arguing that relatively flat solutions yield tight bounds . From an algorithmic perspective , approaches to recover flat minima are still limited . Most notably , Chaudhari et al . ( 2019 ) developed the Entropy-SGD algorithm . Entropy-SGD defines a localentropy-based objective which smoothens the energy landscape based on its local geometry . This in turn allows SGD to attain flatter solutions . Indeed , this approach was motivated by earlier work in statistical physics ( Baldassi et al. , 2015 ; 2016 ) which proves the existence of non-isolated solutions that generalize well in networks with discrete weights . Such non-isolated solutions correspond to flat minima in continuous settings . The authors then propose a set of approaches based on ensembles and replicas of the loss to favor wide solutions . Not too far , recent methods in Bayesian deep learning ( BDL ) have also shown potential to recover flat minima . BDL basically averages over multiple hypotheses weighted by their posterior probabilities ( ensembles being a special case of BDL ( Izmailov et al. , 2018 ) ) . One example , is the stochastic weighted averaging ( SWA ) algorithm proposed by Izmailov et al . ( 2018 ) . SWA simply averages over multiple points along the trajectory of SGD to potentially find flatter solutions compared to SGD . Another example is the SWA-Gaussian ( SWAG ) . SWAG defines a Gaussian posterior approximation over neural network weights . Afterwards , samples are taken from the approximated distribution to perform Bayesian model averaging ( Maddox et al. , 2019 ) . Here we also note the recent work by Patel ( 2017 ) which partially motivates our method . Upon the aformentioned observations in Keskar et al . ( 2017 ) , Patel ( 2017 ) shows that the learning rate lowerbound threshold for the divergence of batch SGD , run on quadratic optimization problems , increases for larger batch-sizes . In general non-convex settings , given a problem with N local minimizers , one can compute N lower bound thresholds for local divergence of batch SGD . The number of minimizers for which batch SGD can converge is non-decreasing in the batch size . This is used to explain the tendency of low-batch SGD to converge to flatter minimizers compared to large-batch SGD . The former result links the choice of batch size and its effect on generalization to the choice of the learning rate . With the latter being a tunable parameter , to our knowledge , developing a dynamic choice of the learning rate that targets convergence to flat minimizers has not been studied before . 2 GENERAL FRAMEWORK . In this paper , we propose a framework that dynamically chooses a Sharpness-Aware Learning Rate to promote convergence to flat minimizers . More specifically , our proposed method locally approximates sharpness at the current iterate and dynamically adjusts the learning rate accordingly . In sharp regions , relatively large learning rates are attained to increase the chance of escaping that region . In contrast , when the current iterate belongs to a flat region , our method returns a relatively small learning rate to guarantee convergence . Our framework can be adopted by any local search descent method and is detailed in Algorithm 1 . Algorithm 1 : Sharpness-Aware Learning Rate ( SALR ) Framework Input : Starting point θ0 , initial learning rate η0 , Number of iterations K. for k = 0 , 1 , . . . , K do Estimate Ŝk , the local sharpness around the current iterate θk ; Set ηk = η0 Ŝk Median { Ŝi } k i=1 ; Compute θk+1 using some local search descent method ( Gradient Descent , Stochastic Gradient Descent , ADAM , . . . ) ; end Return θK . As detailed in Algorithm 1 , at every iterate k , we compute the learning rate as a function of the local sharpness parameters { Ŝk } k i=1 . The main intuition is to have the current learning rate to be an increasing function of the current estimated sharpness . Since the scale of the sharpness at different points can vary when using different networks or datasets ( Dinh et al. , 2017 ) , we normalize our estimated sharpness by dividing by the median of the sharpness of previous iterates . For instance in Figure 2 , despite having a similar sharpness measure , we consider the minimizer around θ = 1 to be sharp relative to the blue plot and flat relative to red plot . Normalization resolves this issue by helping our sharpness measure attain scale invariant properties . One can think of the median of previous sharpness values as a global sharpness parameter the algorithm is trying to learn . When k is sufficiently large , the variation in the global sharpness parameter among different iterates will be minimal . From an algorithmic perspective , SALR exploits a neighborhood around the current iterate to dynamically compute a desired learning rate while simultaneously exploring the sharpness of the landscape to refine this global sharpness parameter . 3 SHARPNESS MEASURE . Several sharpness/flatness measures have been defined in recent literature ( Rangamani et al. , 2019 ; Keskar et al. , 2017 ; Hochreiter & Schmidhuber , 1997 ) . For instance , Hochreiter & Schmidhuber ( 1997 ) computes flatness by measuring the size of the connected region in the parameter space where the objective remains approximately constant . In a more recent paper , Rangamani et al . ( 2019 ) proposed a scale invariant flatness measure based on the quotient manifold . Computing such notions for complex non-convex landscapes can be intractable in practice . In addition to the cited results , Keskar et al . ( 2017 ) quantifies flatness by finding the difference between the maximum value of the loss function within a small neighborhood around a given point and the current value . More specifically , they define sharpness as follows : φ ( ε , θ ) , S ( ε , θ ) 1 + f ( θ ) and S ( ε , θ ) = max θ′∈IBε ( θ ) f ( θ′ ) − f ( θ ) , ( 1 ) where IBε ( θ ) is a euclidean ball with radius ε centered at θ and 1+f ( θ ) is a normalizing coefficient . One drawback of equation 1 is that the sharpness value around a maximizer is nearly zero . To resolve this issue , one can simply modify the sharpness measure in equation 1 as follows : S ( ε , θ ) , max θ′∈IBε ( θ ) f ( θ′ ) − min θ′∈IBε ( θ ) f ( θ′ ) . ( 2 ) It can be easily shown that if θ is a local minimizer , equation 2 is equivalent to equation 1 . Both measures defined in equation 1 and equation 2 require solving a possibly non-convex function which is in general NP-Hard . For computational feasibility , we provide a sharpness approximation by running n1 gradient ascent and n2 gradient descent steps . The resulting solutions are used to approximate the maximization and minimization optimization problems . Here we note that our definition for sharpness does not include a normalizing coefficient , as median { Ŝi } k i=1 in Algorithm 1 plays this role . The details of the approximation are shown in the Definition 1 . Definition 1 . Given θ ∈ Rn , iteration numbers n1 and n2 , and step-size γ , we define the sharpness measure Ŝ ( θ ) , f ( θ ( n2 ) k , + ) − f ( θk ) + f ( θk ) − f ( θ ( n1 ) k , − ) + = f ( θ ( n2 ) k , + ) − f ( θ ( n1 ) k , − ) , where θ ( 0 ) k , + = θ ( 0 ) k , − = θk , θ ( n1 ) k , − = θk − n1−1∑ i=0 γ ∇f ( θ ( i ) k , − ) ∥∥∥∇f ( θ ( i ) k , − ) ∥∥∥ , and θ ( n2 ) k , + = θk + n2−1∑ i=0 γ ∇f ( θ ( i ) k , + ) ∥∥∥∇f ( θ ( i ) k , + ) ∥∥∥ . Remark 2 . In contrast to the measures defined in equation 2 and equation 1 , Definition 1 does not require a ball radius ε . However , our definition requires specifying the step-size γ and the number of ascent and descent iterations . Remark 3 . Running gradient descent/ascent with fixed step-size near a minimizer can return a very small sharpness value even if the minimizer is sharp . This is due to the small gradient norm around a minimizer . To resolve this issue , we normalize the gradient at every descent/ascent step . Moreover , normalizing by the norm of the gradient helps in understanding the radius of the ball containing the iterates { θ ( j ) k , − } n1 j=1 and { θ ( j ) k , + } n2 j=1 . Figure 3 shows the plots of the three different sharpness measures defined in this section when computed for a function f ( θ ) = 0.5θ sin ( 3θ ) + 1 . Notice that the blue plot corresponding to the sharpness measure φ ( · ) attains a zero value at local maximizers compared to a positive value for the other two sharpness plots . Moreover , notice that the sharpness value in these three plots attains a small value near the local minimizer . This can be explained by our choice of radius ε = 0.1 which limits the neighborhood being exploited . Increasing the radius for φ ( ε , · ) and S ( ε , · ) ( increasing n1 and n2 for Ŝ ) will provide higher values around the minimizer . We next show that using our sharpness measure in Definition 1 , gradient descent with SALR framework in Algorithm 1 , denoted as GD-SALR , escapes sharp local minima .

Summary: The paper proposes a simple method (SALR) to encourage the SGD to converge to flatter minima for better generalization. The basic idea is that it increases the learning rate when the sharpness is high, vice versa, such that SGD can escape sharp regions quickly. The sharpness is measured by the difference between the maximum and minimum found by a local SGD with a few fixed steps.