### Searching for

###
**Optimal Per-Coordinate Step-sizes** with

**Victor Sanches Portella**

September 2023

cs.ubc.ca/~victorsp

joint with **Frederik Kunstner**, **Nick Harvey**, and **Mark Schmidt**

**Multidimensional Backtracking**

**Theory Student Seminar @ University of Toronto**

## Gradient Descent and Line Search

### Why first-order optimization?

Training/Fitting a ML model is often cast a **(uncontrained) optimization problem**

Usually in ML, models tend to be BIG

**\(d\) is BIG**

Running time and space **\(O(d)\) **is usually **the most we can afford**

First-order (i.e., gradient based) methods fit the bill

(stochastic even more so)

Usually \(O(d)\) time and space per iteration

### Convex Optimization Setting

**\(f\) is convex**

Not the case with Neural Networks

Still quite useful in theory and practice

More conditions on \(f\) for rates of convergence

**\(L\)-smooth**

**\(\mu\)-strongly convex**

### Gradient Descent

Which step-size \(\alpha\) should we pick?

Condition number

\(\kappa\) Big \(\implies\) hard function

### What Step-Size to Pick?

If we know \(L\), picking \(1/L\) always works

**and is worst-case optimal**

What if we do not know \(L\)?

Locally flat \(\implies\) we can pick bigger step-sizes

If \(f\) is \(L\) smooth, we have

**"Descent Lemma"**

**Idea:** Pick \(\eta\) big and see if the "descent condition" holds

(Locally \(1/\eta\)-smooth)

### Backtracking Line-Search

**Backtracking Line-Search**

Start with \(\alpha_{\max} > 2 L\)

\(\alpha \gets \alpha_{\max}/2\)

**If**

\(t \gets t+1\)

**Else**

**While** \(t \leq T\)

Halve candidate space

**Guarantee**: step-size will be at least \(\tfrac{1}{2} \cdot \tfrac{1}{L}\)

**Armijo Condition**

### Beyond Line-Search?

**Converges in 1 step**

\(P\)

\(O(d)\) space and time \(\implies\) \(P\) diagonal (or sparse)

Can we find a good \(P\) automatically?

**"Adapt to \(f\)"**

**Preconditioer \(P\)**

## "Adaptive" Optimization Methods

### Adaptive and Parameter-Free Methods

Preconditioner at round \(t\)

**AdaGrad from Online Learning**

or

Better guarantees if **functions are easy**

while preserving optimal worst-case guarantees in Online Learning

Attains** linear rate in classical convex opt** (proved later)

But... Online Learning is** too adversarial**, AdaGrad is **"conservative"**

In OL, functions change every iteration **adversarially**

### "Fixing" AdaGrad

But... Online Learning is** too adversarial**, AdaGrad is **"conservative"**

"**Fixes**": Adam, RMSProp, and other workarounds

"AdaGrad inspired anincredible number of clones, most of them withsimilar, worse, or no regret guarantees.(...) Nowadays, [adaptive] seems to denoteany kind of coordinate-wise learning rates that does not guarantee anything in particular."

**Francesco Orabona** in "A Modern Introduction to Online Learning", Sec. 4.3

### Hypergradient Methods

**Idea: **look at step-size/preconditioner choice as an optimization problem

Gradient descent on the hyperparameters

How to pick the step-size of this? Well...

Little/ No theory

Unpredictable

... and popular?!

### Second-order Methods

Newton's method

is usually a great preconditioner

**Superlinear** convergence

...when \(\lVert x_t - x_*\rVert\) small

**Newton **may diverge otherwise

Using step-size with Newton and QN method ensures convergence away from \(x_*\)

**Worse than GD**

\(\nabla^2 f(x)\) is usually expensive to compute

should also help

Quasi-Newton Methods, e.g. BFGS

### State of Affairs

**(Quasi-)Newton**: needs Hessian, can be slower than GD

**Hypergradient methods**: purely heuristic, unstable

**Online Learning Algorithms**: Good but pessimistic theory

at least for smooth optimization it seems pessimistic...

**Online Learning**

**Smooth Optimization**

**1 step-size**

**\(d\) step-sizes**

(diagonal preconditioner )

Backtracking Line-search

Diagonal AdaGrad

Coordinate-wise

Coin Betting

(non-smooth opt?)

**Multidimensional Backtracking**

Scalar AdaGrad

Coin-Betting

**What does it mean for a method to be adaptive?**

## Preconditioner Search

### Optimal (Diagonal) Preconditioner

**Optimal step-size**: biggest that guarantees progress

**Optimal preconditioner**: **biggest (??)** that guarantees progress

**\(L\)-smooth**

**\(\mu\)-strongly convex**

minimizes \(\kappa_*\) such that

**Over diagonal matrices**

### From Line-search to Preconditioner Search

**Line-search**

step-size is at least \(1/2\) the optimum \(1/L\)

# backtracks \(\leq\)

**Multidimensional Backtracking**

Condition number is at least \(1/\sqrt{2d}\) the optimum

# backtracks \(\lesssim\)

**Worth it if \(\sqrt{2d} \kappa_* \ll 2 \kappa\)**

## Multidimensional Backtracking

### Why Naive Search does not Work

**Line-search**: test if step-size \(\alpha_{\max}/2\) makes enough progress:

**Armijo condition**

If this fails, **cut out** everything bigger than \(\alpha_{\max}/2\)

**Preconditioner search:**

Test if preconditioner \(P\) makes enough progress:

Candidate preconditioners \(\mathcal{S}\): diagonals in a box

If this fails, **cut out** everything bigger than \(P\)

### Why Naive Search does not Work

**Preconditioner search:**

Test if preconditioner \(P\) makes enough progress:

Candidate preconditioners \(\mathcal{S}\): diagonals in a box

If this fails, **cut out** everything bigger than \(P\)

### Convexity to the Rescue

\(P\) does not yield **sufficient progress**

Which preconditioners can be thrown out?

All \(Q\) such that \(P \preceq Q\) works, but it is **too weak**

\(P \) does not yield sufficient progress \(\iff\) \(h(P) > 0\)

Convexity \(\implies\)

\(\implies\) \(Q\) **is invalid**

A separating hyperplane!

\(P\) in this half-space

**Hypergradient**

### Convexity to the Rescue

### Box as Feasible Sets

### How Deep to Query?

### Ellipsoid Method to the Rescue

### Smallest Axis-Aligned Ellipsoid

Contraction of \(1/\sqrt{2d}\) from boundary

Constant volume contraction

### Conclusions

Theoretically principled adaptive optimization method for strongly convex smooth optimization

A theoretically-informed use of "hypergradients"

ML Optimization meets Cutting Plane methods

Stochastic case?

Heuristics for non-convex case?

Other cutting-plane methods?

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**Thanks!**

## Backup Slides

#### Preconditioner Search

By Victor Sanches Portella

# Preconditioner Search

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