Online Learning - Algorithms

Outline

• Online Learning Review
• Follow The Leader
• Follow the Regularized Leader
  • Online Gradient Descent
  • Exponntial Weights
  • Extensions
• Applications
  • Electricity Forecasts
  • Ad Click Prediction

Online Learning: Review

• Goal of online learning is to produce sequence of forecasts with low regret
• Each round use past data \(\mathcal{Y}_t=\{y_s\}_{s=1}^{t}\in \otimes_{s=1}^{t} \mathrm{Y}\)
• Apply algorithm \(\mathcal{A}\): sequence of decision rules \(\{\widehat{y}_{t+1}=f_t(\mathcal{Y}_t)\}_{t=1}^{T}\)
• Given a comparison class \(\mathcal{F}_t=\{f_{t,\theta}(): \otimes_{s=1}^{t}\mathrm{Y}\to\mathrm{Y}:\ \theta\in\Theta\}\)
• Want to minimize \(reg_{T}(\mathcal{A})=\underset{\{y_{t+1}\}_{t=1}^{T}\in\otimes_{t=1}^{T}\mathrm{Y}}{\max}\left\{\sum_{t=1}^{T}\ell(y_{t+1},f_t(.))-\underset{\theta\in\Theta}{\min}\sum_{t=1}^{T}\ell(y_{t+1},f_{\theta,t}(.))\right\}\)
• Last class: Prediction With Expert Advice: \(\mathrm{Y}=\{0,1\}\), \(\Theta\) finite, \(\ell()=1\{y_{t+1}\neq\widehat{y}_{t+1}\}\)
  • Saw algorithms (Majority Vote, Hedge) which produce low regret in this setting

General Purpose Algorithms

• In statistical setting, had “generic” methods which take predictor class, loss function and produce a low risk forecast
  • Empirical/Structural Risk Minimization often suffices to achieve near-optimal risk
  • Requires some conditions (stationarity, weak dependence, low complexity)
• For Bayesian setting, similarly had method that, given likelihood and prior, produces a low average risk forecast
  • Bayesian updating and posterior predictive distribution
• Is there a similar general purpose procedure that we can use in online learning setting?
  • Will again require some conditions, and some choices which aren’t automatic
  • Given a loss function and comparison class with certain properties, there is a useful class of methods
• Restricted but highly flexible case: Online Convex Optimization (see e.g. Hazan 2015)
  • Ensures simple methods have strong guarantees

Online Convex Optimization Setting

• Detour from forecasting problem to slightly more general setup
  • Forecasting problem can be shown to be a special case
• Instead of sequence of observations \(y_{t+1}\), face sequence of convex functions \(g_t(x)\in\mathcal{G}\)
  • Convexity means \(g_t(ax+(1-a)x^{\prime})\leq ag_t(x)+(1-a)g_t(x^\prime)\) for any \(a\in[0,1]\)
• Each round, before \(g_t()\) is known, choose \(x_t\) from a set \(\mathcal{K}\)
• \(\mathcal{K}\) is a convex set: if \(x,x^{\prime}\in\mathcal{K}\), then \(ax+(1-a)x^{\prime}\in\mathcal{K}\) for any \(a\in[0,1]\)
• Goal is to minimize regret \(\underset{g_1\ldots g_{T}\in\mathcal{G}}{\max}(\sum_{t=1}^{T}g_t(x_t)-\underset{x\in\mathcal{K}}{\min}\sum_{t=1}^{T}g_t(x))\)
• If function \(g_t\) is the same each round, this is convex optimization setting
  • Goal is to find sequence of values which gets to the bottom of a bowl shaped function
  • Convexity of \(g()\) and \(\mathcal{K}\) means that going downwards always gets to the unique lowest possible value
• In online setting, same principle applies, but now function can change arbitrarily each period

Examples

• In forecasting applications, \(g_t()\) is the loss function each round, and \(\mathcal{K}\) defines space \(\mathcal{F}_t\) of prediction rules
• Online Regression: \(\mathcal{K}=\{\beta\in\mathbb{R}^{n}:\ \Vert\beta\Vert\leq C\}\), \(g_t(\beta_t)=\ell(y_{t+1},\sum_{i=1}^{n}z_{i,t+1}\beta_{i,t})\)
  • Convexity holds so long as \(\ell(.,.)\) convex in second argument: square loss, absolute loss, linear loss, etc
• Other predictor classes may fail to be convex: e.g., general nonlinear functions \(f(z_t,\theta)\)
  • Can allow nonlinearities in predictors \((z_t,z_t^2,z_t^3,\ldots)\), but not usually in parameters
• Prediction with Expert Advice/Hedge
  • Setting with deterministic choices is not convex, but setting with probabilitic strategies is
  • \(\mathcal{K}=\{p\in\mathbb{R}^{n}:\ \sum_{i=1}^{n}p_i=1,p_i\geq 0\}\) \(g_t(p)=\sum_{i=1}^{n}p_{i,t}\ell(y_{t+1},\widehat{y}_{i,t+1})\)
• Improper version: allow weighted average of expert predictions \(\{\widehat{y}_{i}\}_{i=1}^{n}\)
  • \(\mathcal{K}=\{p\in\mathbb{R}^{n}:\ \sum_{i=1}^{n}p_i=1,p_i\geq 0\}\) \(g_t(p)=\ell(y_{t+1},\sum_{i=1}^{n}p_{i,t}\widehat{y}_{i,t+1})\)
  • Assume \(\ell(.,.)\) convex in second argument and bounded
  • Rather than picking one expert at random, take combination of all experts

Follow The Leader

• Goal is to produce low cumulative value \(\sum_{s=1}^{t}g_s(x_s)\) relative to best in class \(\mathcal{K}\) in hindsight
• Why not pick the value \(x_t\) which has performed best so far?
• For each \(t\), Follow the Leader algorithm chooses \(x_{t}=\underset{x\in\mathcal{K}}{\arg\min}\sum_{s=1}^{t-1}g_s(x)\)
• In forecast setting, \(\widehat{\theta}_{t}=\underset{\theta\in\Theta}{\arg\min}\sum_{s=1}^{t-1}\ell(y_{s+1},f_{s,\theta}(.))\)
  • Then predict \(f^{FTL}_{t}(.)=f_{t,\widehat{\theta}_t}(.)\) each round
• Exactly equivalent to running Empirical Risk Minimization over class \(\mathcal{F}_t\) each period
  • This is what you would do in statistical approach, except repeated over and over
• So, does it work? Can we just do what we were already doing and get good results in new setting?
• In general, no
  • For general convex loss functions, regret can grow linearly in \(T\)
  • Means predictor strictly suboptimal every round, no matter how much data you have
• But, in some special cases, yes
  • Return to this after discussing what’s wrong and how to fix it

Worst Case Behavior of Follow The Leader

• Consider case with linear loss \(g_t(x)=g_tx\), \(x\in[-1,1]\)
  • Convexity clearly holds because linear combination of linear functions is linear
• Consider following sequence of \(g_t\) values
  • \(g_1=-0.5\), then
  • \(g_t=1\) if \(t\) even
  • \(g_t=-1\) if \(t>1\) odd
• For odd \(t\), FTL chooses \(x_t=\underset{x\in[-1,1]}{\arg\min} 0.5x\), which is \(-1\)
• For even \(t\), FTL chooses \(x_t=\underset{x\in[-1,1]}{\arg\min}-0.5x\), which is \(1\)
• Loss of FTL therefore \(\propto T\), while loss of best fixed alternative in hindsight, \(x=0\), is 0
• Regret of algorithm is \(\propto T\): average regret is constant: no matter how much data, predictions don’t improve
• Does badly because predictions can be unstable: move a lot between guesses
• At least in this case, suggests looking for algorithms which don’t move so much

Follow the Regularized Leader

• To avoid instability, modify approach to reduce variability of predictions
  • Exactly the purpose of regularization
• Let \(R(x):\ \mathcal{K}\to\mathbb{R}^{+}\) be a convex function, a regularizer
  • Example 1: Euclidean Regularizer \(R(x)=\frac{1}{2}\Vert x\Vert^2\) over \(\mathcal{K}\subseteq\mathbb{R}^n\)
  • Example 2: Entropic Regularizer \(R(x)=\sum_{i=1}^{n}x_i\log x_i\) over \(\mathcal{K}=\{x\in\mathbb{R}^{n}:\ \sum_{i=1}^{n}x_i=1,x_i\geq 0\}\)
• For each \(t\), Follow the Regularized Leader algorithm chooses \(x_{t}=\underset{x\in\mathcal{K}}{\arg\min}(\sum_{s=1}^{t-1}g_s(x)+\frac{1}{\eta}R(x))\)
  • Shortened to FTRL, FoReL or RFTL (for “Regularized Follow the Leader”)
  • Parameter \(\eta>0\) determines strength of the regularization: large values closer to FTL, small values closer to minimizing \(R\)
• In forecast setting, \(\widehat{\theta}_{t}=\underset{\theta\in\Theta}{\arg\min}(\sum_{s=1}^{t-1}\ell(y_{s+1},f_{s,\theta}(.))+R(\theta))\)
  • Then predict \(f^{FTRL}_{t}(.)=f_{t,\widehat{\theta}_t}(.)\) each round
• Corresponds exactly to running Penalized Empirical Risk Minimization each round
• With appropriate choice of \(\eta\) and a few conditions, can guarantee low regret

FTRL Example 1: Online Linear Optimization

• Let’s go back to linear case that gave trouble to FTL algorithm, now in \(\mathbb{R}^{n}\)
• Let \(g_t(x_t)=\sum_{i=1}^ng_{i,t}x_{i,t}\) and use Euclidean/quadratic regularizer \(R(x_t)=\frac{1}{2}\sum_{i=1}^n x_{i,t}^2\)
• FTRL update step is \(x_{t}=\underset{x\in\mathcal{K}}{\arg\min}(\sum_{s=1}^{t-1}\sum_{i=1}^ng_{i,s}x_{i}+\frac{1}{2\eta}\sum_{i=1}^n x_{i}^2)\)
• If constraint that \(x_t\in\mathcal{K}\) not binding (eg if \(\mathcal{K}=\mathbb{R}^n\), or we allow improper choices), solution of this problem gives
  • \(x_1=0\), \(x_{t+1}=-\eta\sum_{s=1}^{t}g_s=x_{t}-\eta g_t\) for any \(t>1\)
  • Update in direction of latest linear direction, by an amount \(\eta\)
• With sufficiently small choice of \(\eta\), fluctuations are damped relative to unregularized case
  • \(\eta\approx \frac{1}{\sqrt{T}}\) gives regret \(\approx\sqrt{T}\) against bad sequence from before
• In fact, can show regret bound of \(\sqrt{2T}\) holds against any sequence of \(g_t\in[-1,1]\) if \(\mathcal{K}=[-1,1]\)
  • Sublinear regret can achieved \(\to\) average regret goes to 0

Linearization Trick

• Previous method presented for linear loss, a single special case
• In fact, linear case is entirely general: same algorithm applies
• The trick is to replace \(g_t(x_t)\) with a linear approximation which provides upper bound on regret
• If \(g_t\) and \(\mathcal{K}\) convex, any subgradient provides a lower bound on the function
  • \(\partial g(x)=\{z\in\mathbb{R}^{n}:\forall u\in\mathcal{K}, g(u)\geq g(x)+\sum_{i=1}^{n}(u_i-x_i)z_i\}\)
  • If \(g_t\) also differentiable, unique subgradient is the gradient \(\nabla g_t(x)\)
• Rearranging, \(g_t(x_t)-g_t(u)\leq \sum_{i=1}^{n}(x_{i,t}-u_{i})z_{i,t}\) \(\forall u\in\mathcal{K}\)
• Summing up \(\sum_{t=1}^{T}g_t(x_t)-\sum_{t=1}^{T}g_t(u)\leq \sum_{t=1}^{T}\sum_{i=1}^{n}x_{i,t}z_{i,t}-\sum_{t=1}^{T}\sum_{i=1}^{n}u_{i}z_{i,t}\)
• In words, this says regret of an online convex optimization problem is less than regret of corresponding online linear optimization problem
• So special case suffices: just use (sub)gradient and solve linear problem

Example Continued: Online Gradient Descent

• Algorithm which takes (sub)gradient of \(g_t(x_t)\) and runs online linear optimization with quadratic penalty is called Online Gradient Descent (OGD)
• Starting with \(x_1=0\), each round suffer loss \(g_t(x_t)\), and calculate subgradient \(\nabla g_t(x_t)\in\partial g_t(x_t)\)
  • Then predict \(x_{t+1}=\underset{x}{\arg\min}(\sum_{s=1}^{t}\sum_{i=1}^{n}\nabla g_{i,t}x_{it}+\frac{1}{2\eta}\sum_{i=1}^{n}x_{i,t}^2)\)
• Update step simplifies to \(x_{t+1}=x_t-\eta\nabla g_{t}\)
  • This is exactly gradient descent algorithm used for optimization
• Regret guarantee (cf Shalev-Shwartz Cor. 2.7): suppose \(\underset{t}{\max}\Vert\nabla g_{t}\Vert^2\leq L\) and \(\mathcal{K}\subseteq\{x\in\mathbb{R}^{n}:\ \Vert x\Vert\leq B\}\)
  • Bound on subgradients is a smoothness condition, bound on \(\mathcal{K}\) restricts comparison class
  • Result: if \(\eta=\frac{B}{L^2\sqrt{2T}}\), obtain \(Reg_{T}(OGD)\leq BL^2\sqrt{2T}\)
• Application: Online Regression: Consider \(g(\beta_t)=\ell(y_{t+1},z_{t+1}^{\prime}\beta_t)\)
  • Corresponds to \(L_2\) penalized regression each period
  • Update rule is \(\beta_1=0\), \(\beta_{t+1}=\beta_t-\eta \frac{d}{d\widehat{y}}\ell(y_{t+1},z_{t+1}^{\prime}\beta)z_t\)
  • If \(z_{t+1}\) bounded, subgradient bound holds for absolute loss or logistic regression
    • Not for square loss: needs another approach

FTRL Example 2: Exponential Weights Methods

• For cases where decision is a probability distribution, prefer method which stays in space of distributions
  • \(\mathcal{K}=\{x\in\mathbb{R}^{n}:\ \sum_{i=1}^{n}x_i=1,x_i\geq 0\}\) is probability simplex
  • Can stay in \(\mathcal{K}\) using entropic penalization: \(R(x)=\sum_{i=1}^{n}x_{i}\log x_i\)
• FTRL with linear loss and Entropic Penalty gives a distribution over predictions
  • \(x_{t}=\underset{x\in\mathcal{K}}{\arg\min}(\sum_{s=1}^{t}\sum_{i=1}^{n}g_{i,s}x_{i}+\frac{1}{\eta}\sum_{i=1}^{n}x_{i}\log x_i)\)
• With some clever algebra, can show form of update is \(x_{i,t+1}=\frac{x_{i,t}e^{-\eta g_{i,t}}}{\sum_{j=1}^{n}x_{j,t}e^{-\eta g_{j,t}}}\) for any \(i=1\ldots n\)
  • This is exactly Hedge algorithm: update probability of choice \(i\) in proportion to exponentiated time \(t\) loss
• Apply linearization trick for nonlinear objectives: replace \(g_{i,t}\) with (sub)gradient \(\nabla g_{i,t}\)
  • Algorithm called (Normalized) Exponentiated Gradient
• If \(\underset{i,t}{\max}|\nabla g_{i,t}|\leq G\) and \(\eta=\sqrt{\frac{\log n}{2TG^2}}\) regret bound (cf Hazan cor. 5.5)
  • \(Reg_{T}\leq 2G\sqrt{2T\log n}\): sublinear, logarithmic dependence on number of experts
• Application: weighted averages, \(g_t():=\ell(y_{t+1},\sum_{i=1}^{n}p_{i,t}\widehat{y}_{i,t+1})\)
  • \(\nabla g_{i,t}=\frac{\partial}{d\widehat{y}_{t+1}}\ell(y_{t+1},\sum_{i=1}^{n}p_{i,t}\widehat{y}_{i,t+1})y_{i,t+1}\)
  • Exponentiated gradient method provides combination of any set of predictions with bounded prediction, smooth loss

General Results (Shalev-Shwartz Thm 2.11/Hazan Thm 5.1)

• Under boundedness and smoothness, with proper choice of \(\eta\), two special cases of FTRL produce regret \(\leq C\sqrt{T}\)
• Is this a general phenomenon? Yes, under some conditions
  • Regularizer \(R(x)\) is strongly convex: \(R(x)\) bounded below by quadratic function of \(\sigma\Vert x \Vert^2\) for some norm
  • Losses are smooth (Lipschitz) \(\Vert\nabla g_t(x)\Vert\leq L<\infty\) in same norm
  • Set \(\mathcal{K}\) is bounded \(\underset{x\in\mathcal{K}}{\max}(R(x)-\underset{u\in\mathcal{K}}{\min}R(u))=B<\infty\)
    
• Then FTRL with \(\eta=\frac{B}{L^2\sqrt{2T}}\) has regret \(\leq BL^2\sqrt{2T}\)
• So, if objective convex, you can find effective regularizer, and model is smooth, FTRL has low regret
  • Rules out many problems where domain completely unrestricted
    • E.g., in financial forecasting, one day’s losses may wipe out years of gains
    • Huge \(B\) makes bounds useless
  • Regret primarily meaningful if comparison class likely to perform well
    • Trade off low regret vs good performance of comparison class
• For this class of problems, FTRL is general-purpose method, in same way ERM/SRM is in statistical case

Revisiting Follow the Leader

• Given good performance of FTRL, bad worst case for Follow the Leader, should we always be penalizing?
• In some cases, Follow the Leader does have strong guarantees
  • If \(g_t(x_t)\) \(\sigma\)-strongly convex, FTL can perform as well or better than FTRL
• Intuition: By linearization, \(\sum_{s=1}^{t}g_s(x_t)-g_s(u)=\sum_{s=1}^{t}\nabla g_s^{\prime}(x_s-u)+\sum_{s=1}^{t}(g_s(x_s)-\nabla g_s^{\prime}(x_s-u))\)
  • Nonlinear strictly convex part acts like regularizer to linear part
  • So unpenalized method equivalent to (some kind of) penalized method
• Knowing that \(g_t(x_t)\) strongly convex can also buy improved regret bounds
  • May obtain \(Reg_{T}\propto \log (T)\) instead of \(\sqrt{T}\)
  • Very substantial accuracy improvement in terms of amount of data needed
• For Online Gradient Descent with strictly convex \(g_t()\), can run \(x_{t+1}=x_{t}-\eta_{t}\nabla_t g_t\)
  • If \(\eta_t=\frac{1}{\sigma t}\), and \(\Vert \nabla g_t \Vert \leq G\), regret bound is \(\frac{G^2}{2\sigma}(1+\log T)\)
  • Bound \(\infty\) in non-strong case \(\sigma=0\)
  • But if known to be strong, can take bigger steps at start, smaller ones at end
• Cases with strong convexity, like many online regression problems, can go faster than linear problems like Hedge

Extensions and Adaptivity

• In practice, hard part of implementation is choosing scaling parameter \(\eta\)
• Need to know time frame \(T\), bounds \(B\), and smoothness level \(L\)
• Usual tricks, like cross-validation, not usually possible in an online setting
• This has led to variety of modified algorithms which choose \(\eta\) adaptively
  • Penalization level depends on data, and may change at each \(t\)
• If time frame not known, simple approach is doubling trick
  • Start as if \(T_0\) fixed, then restart when reached and set \(T_{k+1}=2T_{k}\)
• Can also use continuously-updated penalties, e.g. \(\eta_t=\frac{1}{\sqrt{t}}\)
  
• If smoothness not known, popular method is Adagrad
  • Change Online Gradient Descent update to \(x_{i,t+1}=x_{i,t}-\frac{\eta}{\sqrt{\sum_{s=1}^{t}\nabla g_{i,s}^2}}\nabla g_{i,t}\)
  • Updates slower for big gradients, smaller for large ones
• For online gradient descent, can enforce choices in \(\mathcal{K}\): called “Projected Online Gradient Descent”
  • After each update, move to closest point in \(\mathcal{K}\) to \(x_{t+1}\) after each gradient step

library(opera) #Library for online learning
library(mgcv) #Library for additive models: used as part of the expert forecasts
library(caret) #Library for several machine learning tasks: used as part of the expert forecasts

set.seed(1) #Ensure randomness always identical (only shows up in GBM model)

data(electric_load)
idx_data_test <- 620:nrow(electric_load)
data_train <- electric_load[-idx_data_test, ] 
data_test <- electric_load[idx_data_test, ]  

#Expert 1: A Generalized Additive Model, in Several Predictors
gam.fit <- gam(Load ~ s(IPI) + s(Temp) + s(Time, k=3) + 
                s(Load1) + as.factor(NumWeek), data = data_train)
gam.fcst <- predict(gam.fit, newdata = data_test)

#Expert 2: "medium term model", which adds autoregression component on residuals of an additive model
medium.fit <- gam(Load ~ s(Time,k=3) + s(NumWeek) + s(Temp) + s(IPI), data = data_train)
electric_load$Medium <- c(predict(medium.fit), predict(medium.fit, newdata = data_test))
electric_load$Residuals <- electric_load$Load - electric_load$Medium

# autoregressive correction
ar.fcst <- numeric(length(idx_data_test))
for (i in seq(idx_data_test)) {
  ar.fit <- ar(electric_load$Residuals[1:(idx_data_test[i] - 1)])
  ar.fcst[i] <- as.numeric(predict(ar.fit)$pred) + electric_load$Medium[idx_data_test[i]]
}

# Expert 3: A Gradient Boosting model (a machine learning thing based on trees) 
capture.output(gbm.fit <- train(Load ~ IPI + IPI_CVS + Temp + Temp1 + Time + Load1 + NumWeek, 
                  data = data_train, method = "gbm"),file="/dev/null")
#capture.output is used to prevent command from spewing hundreds of lines of text
gbm.fcst <- predict(gbm.fit, newdata = data_test)


# Combine expert forecasts into sequences X along with observed outcomes Y
Y <- data_test$Load
X <- cbind(gam.fcst, ar.fcst, gbm.fcst)

#Find loss of best expert ex post, according to absolute loss, and compare individual experts
oracle.expert<-oracle(Y = Y, experts = X, loss.type = "absolute", model = "expert")

params<-list(alpha=0.5,simplex=TRUE) # Code uses \eta=t^_{\alpha}, and projects to simplex
#Select weighting of experts by projected online gradient descent
ogdexperts<- mixture(Y=Y, experts=X, model = "OGD", loss.type = "absolute",parameters = params)

param2<-list(eta=0.5)
#Select weighting of experts by exponentially weighted average
exponentialexperts<- mixture(Y=Y, experts=X, model = "EWA", loss.type = "absolute",parameters = param2)

Application: Electricity Forecasting

• Several prediction with expert advice algorithms implemented in R in library opera
• Toy example: predict weekly electricity consumption
  • Task important for power companies, which must buy and sell excess production on interchange markets
• Train 3 complicated statistical/machine learning models on training set, then every week use their forecasts with incoming predictors (tempurature, season, etc) to predict that week’s usage
  • Label as gam.fcst, ar.fcst, and gbm.fcst
• Best expert of 3 ex post, by absolute loss over 112 weeks, is gam.fcst with loss of 1122.4255307
• Predict online using weighted average of 3 experts, by exponential weights and projected Online Gradient Descent
  • Let \(\eta=0.5\) for exponential weights, and \(\eta_t=t^{-0.5}\) for OGD
• Results: Exponential weights achieves regret 23.1039162, and OGD regret 6.817474
  • Both perform nearly as well as best expert ex post, even though not known ex ante

Projected Online Gradient Descent Results

#Plot results
plot(ogdexperts)

Exponential Weights Results

#Plot results
plot(exponentialexperts)

Application: Ad-Click Prediction at Google

• Google implemented system to forecast probability of clicking on ads (McMahan et al 2013)
• Want system to automatically give new prediction for each ad, for each customer
  • Apply online method to update continuously and automatically
• Want to use ad and user-level attributes for prediction
  • Apply features in online logistic regression \(g_t(\beta_t)=-y_t\log(\frac{1}{1+\exp(-z_t^\prime\beta_t)})-(1-y_t)\log(\frac{\exp(-z_t^\prime\beta_t)}{1+\exp(-z_t^\prime\beta_t)})\)
• \(T\) is in the billions per day, and \(z_t\) has billions of dimensions
  • Individual sites and each have own attributes (ie, each person’s search history, etc)
  • Need extreme speed and scale: use sparse prediction method, using only a few coefficients at a time
• Approach: Linearized FTRL with particular choice of regularizer
  • \(\beta_{t+1}=\underset{\beta}{\arg\min}(\sum_{s=1}^{t}\nabla g_s\beta+\sum_{s=1}^{t}\sigma_s\Vert\beta-\beta_s\Vert^2+\lambda_1\Vert \beta\Vert_1)\)
  • Uses regularizer depending on whole past sequence of \(\beta_s\), plus L1 penalty
  • Latter acts like Lasso penalty; former like Euclidean, but leads to more computationally efficient updates
• Online methods also used at Yahoo, Microsoft, many other web companies
  • Vowpal Wabbit system implements fast methods for large-scale web applications
  • Speed, flexibility, reliability provided by online learning approach

Conclusions

• Online Learning posesses simple, reliable general purpose classes of algorithms
• If objective is convex and smooth, and predictor sets are bounded, can rely on Follow The Regularized Leader
  • Applies penalized risk minimization each period
  • If penalty chosen to balance out these factors, regret \(\propto \sqrt{T}\)
• Special Case: Online Gradient Descent: update parameters in direction of subgradient
• Special case: Exponential Weights: update probabilities in proportion to exponent of loss
• Follow the Leader (Empirical Risk Minimization each round) less universal, but can be very accurate in strongly convex case

References

• Elad Hazan Introduction to Online Convex Optimization Foundations and Trends in Optimization, vol. 2, no. 3-4 (2015) 157–325.
• Shai Shalev-Shwartz Online Learning and Online Convex Optimization Foundations and Trends in Machine Learning Vol. 4, No. 2 (2011) 107–194
  
• H. Brendan McMahan et. al. Ad Click Prediction: a View from the Trenches KDD’13 (2013)
  • Description of Google’s online-learning system