Tutorial on Machine Learning for Advanced Electronics

Transcription

1 Tutorial on Machine Learning for Advanced Electronics Maxim Raginsky March 2017

2 Part I (Some) Theory and Principles

3 Machine Learning: estimation of dependencies from empirical data (V. Vapnik) enabling a computer to perform well on a given task without explicitly programming the task improving performance on a task based on experience

4 Statistical Learning Paradigm Y }{{} output = f }{{} ( ) }{{} X input Experience comes in the form of training data (X 1, Y 1 ), (X 2, Y 2 ),..., (X n, Y n ) sampled at random from the phenomenon of interest unknown distribution p(x, y) Performance is measured by expected accuracy of inference (prediction on previously unseen instances): R(f) = E[l(f(X), Y )] = l(f(x), y) }{{} prediction error p(x, y) }{{} data distribution dx dy Improvement comes from seeing more data before settling on a hypothesis

5 Goals of Learning (X 1, Y 1 ),..., (X n, Y n ) }{{} training data Predictive consistency algorithm R( f n ) n min f F R(f) f n }{{} hypothesis eventually, learn to predict as well as the best predictor you could find with full knowledge of p(x, y) Generalization 1 n n l( f n (X i ), Y i ) R( f n ) i=1 performance of f n on the training data is a good predictor of its performance on previously unseen instances Efficiency minimize resource usage (data, computation time, memory,...)

6 Learning and Inference Pipeline Learning and inference pipeline Learning Training Samples Features Training Labels Training Learned model Inference Learned model Features Predic'on Test Sample Image credit: Svetlana Lazebnik

7 Data Requirements Predictive consistency: for given ε, δ > 0, what is the smallest n, such that with probability 1 δ? R( f n ) min f F R(f) ε Generalization: for given ε, δ > 0, what is the smallest n, such that 1 n l( n f n (X i ), Y i ) R( f n ) ε i=1 with probability 1 δ? Efficiency the golden standard is n(ε, δ) 1 ε 2 log 1 δ from P[error diff. ε] c 1 e c 2nε 2

8 Underfitting and Overfitting Underfitting: not enough data R( f n ) min R(f) is large f F Overfitting: low training error but large testing error R( f n ) 1 n l( n f n (X i ), Y i ) is large i=1 High Bias Low Variance Low Bias High Variance Prediction Error ts Test Sample Training Sample Low Model Complexity High Image credit: Hastie et al., Elements of Statistical Learning

9 Generative and Discriminative Models (X 1, Y 1 ),..., (X n, Y n ) }{{} training data algorithm f n }{{} hypothesis Recall: distribution p(x, y) is unknown Generative modeling: use the data to estimate p, then find the best predictor for p n : (X 1, Y 1 ),..., (X n, Y n ) p n (x, y) f n terminology: generative model explicitly models how input x and the output y are generated advantages: pn can be used for other tasks besides prediction (e.g., exploring the data space) disadvantages: inflated sample complexity, especially in high dimension

10 Generative and Discriminative Models (X 1, Y 1 ),..., (X n, Y n ) }{{} training data algorithm f n }{{} hypothesis Recall: distribution p(x, y) is unknown Discriminative modeling: focus on prediction only (X 1, Y 1 ),..., (X n, Y n ) f n terminology: discriminative model (implicitly) models the distribution of y for each fixed x, allowing to discriminate between different values of y advantages: smaller data requirements disadvantages: narrowly focused; will need retraining if the task changes

11 Part II Canonical ML Tasks and Algorithms

12 Supervised and Unsupervised Learning Prediction is an example of supervised learning: training data consist of input instances matched (or labeled) with output instances to guide the learning process Many learning tasks are unsupervised: no labels provided, and the goal is to learn some sort of structure in the data Performance criteria are more vague and subjective than in supervised learning Unsupervised learning is often an initial step in a more complicated learning and inference pipeline

13 Unsupervised Learning Clustering: Unsupervised discoverlearning groups of similar instances Clustering Discover groups of similar data points Manifold learning: discover low-dimensional structures Density estimation: approximate probability density of data Unsupervised learning is often used at early stages of ML pipeline. Image credits: Svetlana Lazebnik

14 Supervised Learning: Classification (X, Y ) p input/output Classification: assign attributes X to one of M classes Example: spam detection Criterion: probability of misclassification R(f) = P[f(X) Y ], so that l(y, f(x)) = { 0, if f(x) = Y 1, if f(x) Y

15 Consider the following setting. Let Supervised Learning: Regression (X, Y ) p Y = f (X) + W, where X is a random variable (r.v.) on X = [0, 1], W is a r.v. on Y = R, independent of X input/output Finally let f : [0, 1] R be a function satisfying E[W ] = 0 and E[W 2 ] = σ 2 <. f (t) f (s) L t s, t, s [0, 1], Regression: predict a continuous output Y given X Example: curve fitting from noisy data where L > 0 is a constant. A function satisfying condition (1) is said to be Lipschitz on [0, such a function must be continuous, but it is not necessarily differentiable. An example of is depicted in Figure 1(a) (a) (b) Figure 1: Example of a Lipschitz function, and our observations setting. (a) random samp points correspond to (X i, Y i ), i = 1,..., n; (b) deterministic sampling of f, the points (i/n, Y i ), i = 1,..., n. Criterion: mean squared error R(f) = E[(f(X) Y ) 2 ], so that l(y, f(x)) = (f(x) Y ) 2 Note that

16 Example 1: Kernel Methods Training data: (X 1, Y 1 ),..., (X n, Y n ) Look for f n of the form n f c (x) = c i K(x X i ) i=1 Coefficients c 1,..., c n chosen to minimize n (f c (X i ) Y i ) 2 i=1 (subject to additional constraints) K(x x ): potential field due to unit charge at x Advantage: easy to optimize Disadvantage: need to store c and X i s to do inference Image credit: Aizerman, Braverman, Rozonoer

17 Example 2: Neural Networks and Deep Learning x 1 w 1 x 2 x d. w 2 w d 1 b y = f(w 1 x w d x d + b) where: w 1,..., w d tunable gains b tunable bias f( ) activation function f y Build up complicated functions by composing simple units Examples of activation functions: { 1, z 0 threshold, f(z) = 0, z < 0 sigmoid, f(z) = 1 1+e z tanh, f(z) = ez e z e z +e z rectified linear unit (ReLU), f(z) = max(z, 0) Advantage: neural networks are universal approximators for smooth functions y = f(x)

18 Example 2: Neural Networks and Deep Learning z 1 = f(a 1 x + b 1 ), z 2 = f(a 2 z 1 + b 2 ),..., y = f(a k z k 1 + b k ) Advantages: deep networks can efficiently represent rich classes of functions; work extremely well in practice Disadvantages: training is computationally intensive; poorly understood theoretically

19 Part III Interaction between Data and Algorithms

20 Recap: Learning and Inference Pipeline Learning and inference pipeline Learning Training Samples Features Training Labels Training Learned model Inference Learned model Features Predic'on Test Sample What happens before the training step?

21 Feature Extraction and Dimensionality Reduction Which features to use? traditionally, this step relied on extensive expert knowledge hand-engineered features tailored to the task at hand deep learning aims to automate this task (think of each layer performing feature extraction) Dimensionality reduction use unsupervised learning to discover low-dimensional structure in the data for example, principal component analysis (PCA)

22 Principal Component Analysis (PCA) compute sample covariance matrix: Σ n = 1 n n (X i µ n )(X i µ n ) T, µ n = 1 n X i n i=1 }{{} sample mean i=1 eigendecomposition: Σ n = UDU T retain top k eigenvectors Image source: MathWorks

23 What To Do If You Don t Have Enough Data? Bootstrap Training data: (X 1, Y 1 ),..., (X n, Y n ) If n is small, there is risk of underfitting Bootstrap is a resampling method: for j = 1 to m end for draw I(j) uniformly at random from {1,..., n} let Z j = (X I(j), Y I(j) ) return Z 1, Z 2,..., Z m Bootstrap can be used to get a good estimate of error bars for a chosen class of models

24 Gradient Descent and Its Relatives Training is an optimization problem: minimize L(w) = 1 n n (Y i f(w, X i )) 2 e.g., x f(w, x) is a deep neural network, so w is a (very) high-dimensional vector i=1 Gradient descent w t+1 = w t η t L(w t ), η t learning rate Backpropagation: use chain rule to compute gradients efficiently Computing L(w) using backprop takes O(nkm) operations, where k is the number of layers and m is the number of connections per layer Still computationally expensive, requires full pass through training data at each iteration

25 Stochastic Gradient Descent Gradient descent w t+1 = w t η t L(w t ), η t learning rate L(w) = 1 n w (Y i f(w, X i )) 2 n i=1 Stochastic gradient with mini-batches: essentially, bootstrap for t = 1, 2,... draw I(1, t),..., I(b, t) with replacement from {1,..., n} w t+1 = w t η t 1 b w l(y I(j,t) f(w t, X I(j,t) )) 2 b end for j=1 Computational-statistical trade-offs: each iteration takes O(bkm) operations instead of O(nkm) gradient error is now 1/b instead of 1/n

26 Part IV Machine Learning for EDA

27 Main Challenges for EDA Training data are waveforms X = (X(t)) 0 t T, Y = (Y (t)) 0 t T e.g., X is voltage, Y is current Many different types of models/analysis: steady-state transient small-signal... How do we represent waveform data? sampling rate affects quality of learned models different features are useful for different analyses (rise/hold time; frequency/phase; duration;...) How can we optimize measurement collection process?

28 Recurrent Neural Networks y 1 y 2 y 3 y T 0 1 f f 2 3 f... T 1 f T x 1 x 2 x 3 x T Nonlinear state-space model: ξ t = g(aξ t 1 + Bx t ) y t = h(cξ t + Dx t ) Physics-based models of nonlinear electronic devices are state-space models RNNs are universal approximators for such systems States (internal variables) can be discovered automatically, without hand tuning

29 Learning RNN Models of Nonlinear Devices y 1 y 2 y 3 y T 0 1 f f 2 3 f... T 1 f T x 1 x 2 x 3 x T Data: sampled waveforms (τ - sampling period, T - # samples) X i = (X i (0), X i (τ),..., X i (T τ)) Y i = (Y i (0), Y i (τ),..., Y i (T τ)) i = 1,..., n RNN is a T -layer NN, with parameters shared by all neurons Training by SGD with Backpropagation Through Time (BPTT) Output: weight matrices A, B, C, D; discrete-time approximation ξ k+1 = g(aξ k + Bx k ) y k = h(cξ k + Dx k )

30 Main Challenges Which stimuli to use? The learned model may not be equally suitable for different types of analysis (transient vs. steady-state) The learned model should be stable, plug-and-play with Verilog-A Enforcing physical constraints: zero in/zero out, symmetry, polarity, etc. Usual issues with BPTT: vanishing/exploding gradients, etc.

31 Capturing Variability: Generative Modeling Device model: y = g(x, β), where x is input, y is output, β are device parameters. Variability: β is a random quantity which varies device to device, yet cannot be sampled directly Data: sample input-ouptut behavior from many devices of the same type Learn a generative model for (x, y) use deep NN to learn complicated functions mapping explicit design parameters to latent variability factors maximum likelihood via SGD with backprop bootstrap can be used to compensate for lack of data Disadvantage: training takes time Advantage: once learned, model is easy to sample from, can be used to explore variability, 3σ typical behavior, etc.

32 Compositional ML Training and inference/simulation pipeline: { } training ξ = f(ξ, x) Verilog-A data simulation y = g(ξ, x) Theoretical and practical challenge: optimize end-to-end performance The learned model should be internally stable enforce stability during training via appropriate penalty functions The learned model should be ready for Verilog-A RNN outputs a discrete-time model Verilog-A uses adaptive time discretization to simulate behavior need deep understanding of numerical methods for ODEs to ensure end-to-end stability and robustness

33 For more questions: