prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How can players skip dialogue on the quest map?

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Context 1: Hey folks! Here is the shiny new Changelog thread  We're including the archived patch notes from the old forums, so that they are preserved for anyone that would like to reference back to them  We will continue to update this thread with new notes as the patches are released 
Chat is now accessible from the quest board, upgrade screen, and many other menus 
Tapping on objects and menus may reveal helpful hints about that object 
Team PI is now colored red if lower than recommended for the quest 
Many text fixes and consistency improvements 
• A new Basic Catalyst found in Special Events is used in every recipe!
Several heroes have received improvements to their base stats 
The abilities of all Champions have increased in effectiveness 
A new Critical Boost buff has been introduced 
Iron Fist and Spiderman now have the ability to Armor Break with their Critical Hits 
Deadpool’s ability to Regenerate is more powerful, but only triggers once per fight 
Scarlet Witch now has a chance to trigger Nullify off of any Critical Hit 
Juggernaut and Rhino now have a layer of Armor 
Punisher and Winter Solider now may also trigger Fury in addition to Bleed 
Colossus now further increases his base Armor with the Armor Up ability 
Thor and Ronan no longer Armor Break; instead, base stats and Stun durations have improved 
We reduced the effectiveness of the Revive items in order to give away more as rewards 
A bonus of 50% for using ISO-8 matching your Champion’s Class can now be previewed on the Upgrade screen 
It’s now possible sell Champions in exchange for ISO-8 and Gold  The amount received increases proportionately to the Rank and Level of the sold Champion 
-You can now skip dialogue on the quest map by pressing ‘SKIP’ 
-Added a ‘Quit’ button directly on the quest interface 
-The Back button on the Top Bar now returns the player to the Home screen 
-Various game balance and cosmetic improvements to the available quests 
• PVP energy has been replaced with Hero Stamina  Each Hero has their own Stamina values, meaning the more Heroes you have the more you can play in PVP 
• Each Hero has 1 Stamina and takes 2 hours to recharge 
We have removed the Next Quest button for a much more favorable and flavorful approach to teaching and informing people about Marvel : Contest of Champions  In the Main Menu(Bottom Right Corner) you will now see an image of the Collector showing you what the best or recommended actions that you should preform  This can be anything from opening Crystals, Continuing a Quest, Ranking Up Champions if the difficulty is too hard, Tips where to obtain items, and Playing Versus/Arenas 
• Adjusted the PI calculation for Power Burn and Power Drain abilities to improve accuracy 
• Significantly increased the Power Burn multiplier as well as the amount of Power burned  Prior to this change, Vision's Special Attack damage output was far below the curve  Vision's Special Damage is distinct from other heroes in that the dependency on opponents' Power levels cause the damage dealt to be highly variable, and sometimes quite low; however, when striking an opponent with high Power levels, Vision has the potential to deal very high amounts of direct, Armor-ignoring damage 
• Slightly adjusted the Armor Break trigger to be less punishing to opponents with the Armor Up ability without sacrificing PI or damage output 
• Slightly increased his base Health and, in turn, the amount of Health recovered by Regeneration  This improvement is reflected by an increase to PI of about 1% 
• Slightly reduced the damage from Bleeding, but slightly increased the amount of Power drained by E M P  Arrow to compensate  This added utility strengthens the choice between whether to offensively Bleed the enemy or defensively drain their Power  These changes may modify PI by +/-1% 
• Slightly reduced the frequency of Nullify for basic attacks, but slightly increased the chance a Special Attack is critical  Chaotic Bombardment no longer has a chance to critical, and instead has a 100% chance to Nullify the target  This is less punishing to opponents with beneficial effects, while providing a more reliable source of Nullify  Overall, her PI has decreased by about 2% 
• Decreased base Health and Attack by 2% each to bring his PI in line with other Champions without compromising Special Attack effectiveness 
• Slightly increased base Health by 2% to bring his PI in line with other Champions  This change may result in a PI increase of up to 1% 
• Fixed a bug with her Bleed ability scaling incorrectly  This has no effect on PI 
• User's on iPhone 4 devices will no longer encounter a progression blocker after fighting Iron Man in the tutorial 
• Fixed an issue where player's Hero would disappear after using a special move 
• Fixed an issue where very rarely a character would lose all functionality when dashing 
• Added additional Network support to better diagnose disconnects  The game should resolve and recover much more gracefully than in previous updates 
• Adjusted some of the touch sensitivity while fighting  Heroes moves should feel more responsive  This is something that is going to be an ongoing process  Please let us know how you think it feels 
• Fixed various issues with Chat 
• We have updated Open GL versions/drivers for iOS devices that support Open GL 3 0 
• User's will no longer receive delayed Game Center notifications  This caused some weirdness to occur while opening Crystals in the Crystal Vault 
• The Crystal Vault has received another polish pass and should now feel much more responsive, thank you for all your feedback on this feature!
• Many more minor bug fixes were included in this update 
• Special Attack 1 base damage increased by +25% Attack Rating 
• Heavy Attack base Power gained reduced to 63 points 
We recently improved the functionality of Heavy Attacks, so they’re easier to use  Their base Power has been reduced to normal levels – previously, they generated Power at a higher rate to compensate for their difficult execution  Special Attacks have been adjusted to give the unlucky recipients more of a fighting chance  These changes bring these attacks in line with existing damage-to-power ratios 
*NOTE: Special Attacks only generate Power for the target struck, not for the user; this prevents infinite loops and helps serve as a comeback mechanic 
Versus Crystal prizes have been adjusted due to the Champion Stamina changes 
Arena Crystal prizes have been increased to help balance the adjustments to the Versus Crystal 
Payouts have significantly increased when receiving a duplicate Champion with a Star rating of two or more  The boosted amount increases based on Star rating  We apologize for any inconvenience caused by delivering each reward individually, and are working to get a fix to you as soon as possible  In the meantime, using the “Skip” button avoids the inconvenience 
• We fixed a bug where finding a new match could cost a player Units 
• Spending Units to find a new opponent will now return opponents with lower ratings 
• Chapters 3 and 4 of Act 2 Story Quests are now available  A mysterious opponent awaits you at the end of Act 2!
*NOTE: This caused some players' progress to reset for a brief time, but that issue should now be corrected 
• Event Quest difficulty has been adjusted to match Catalyst availability 
• Rank-Up Recipes have been adjusted to be more accessible across all ranks 
• Bosses for the Monday through Saturday Daily Events now have a small chance to drop a Class Catalyst  This is in addition to the drop chance from Chests 
• Ambush Rates have been adjusted on all Event Quests 
• Increased Catalyst drops for the Collector Free-For-All Event Quest 
• Alpha Catalysts now have a chance to drop from Chests in Medium and Hard difficulties of The Collector Free-For-All event 
• The unobtainable chest in Act 1, Chapter 1, Quest 6 has been removed from the Battlerealm 
Increased the amount of Gold awarded by the Arena Crystal 
Slightly reduced the cost to level-up a 3-Star Champion at Rank 1 to cleanly align with ISO-8 chunk values 
Fixed a bug with Billion-Dollar Punch not triggering Armor Break 
• Duplicate 2-Star, 3-Star, and 4-Star Champions now awaken a brand new ability unique to that Champion in addition to the rare ISO8 they currently give  Duplicates thereafter continue to level-up this ability to make it stronger  When a Champion is awakened, their Stars turn bright and glow, making them easy to identify (and look pretty cool too)  These new abilities can be quite powerful, so please fight responsibly!
• Various other improvements, including rank and level information for opponents, find match options in team select, and animation tuning 
• There is now a chance to encounter the elusive Treasure Adaptoid, who divulges his hoard of ISO8 and Gold to those able to defeat him in battle 
• Class Relationships can be viewed by tapping “Enemy Classes” before entering a quest, and preview the number of enemies in that quest for each class type 
• You can also now see rewards for completion and exploration on the Edit Team screen 
• Opponents are more aware of the distance between you and them, improving their interaction with knockback effects, such as that from Heavy Attacks 
Mutant Champions are now effective against Skill Champions 
• The high Special Attack damage and regenerative abilities of Mutant Champions are effective against Skill Champions, which typically rely on Bleed damage from their weaponry  We think of this relationship as if the X-Gene grants Mutant Champions superpowers that evolved to be stronger than Champions that are merely “Skilled” 
Skill Champions are now effective against Science Champions 
• While scientists fiddle in their cute little laboratories to create flasks full of serums to turn even frail young men into super-soldiers, Skill Champions were just born that way baby  Often donning sharp weaponry to make their opponents Bleed, Skill Champions enjoy watching the high base attributes of Science Champions just melt away 
Cosmic Champions are now effective against Tech Champions 
• Tech Champions construct durable robots and thick suits of Armor to outlast their opponents in battles of tank-the-nuke   which gives Cosmic Champions extra time to build up stacks of beneficial effects to overrun Tech Champions using their peculiar alien enhancements 
• Tech Champions are still effective against Mutant Champions 
Tech Champions typically excel at Armor, Resistance, and Power manipulation, which is effective against the high Special Attack damage of Mutant Champions  Think of the robotic Sentinels adapting for tactical advantages in the war against Mutantkind!
Science Champions are still effective against Mystic Champions 
• Science Champions – a Class of behemoths like Hulk and super-soldiers like Captain America – typically have above average base attributes like Health, Attack, and Armor  These raw stats cannot be affected by pesky Mystics and their removal abilities: Nullify and Purge 
Mystic Champions are still effective against Cosmic Champions 
• Cosmic Champions explore strange new beneficial effects to seek out new power and new abilities, to boldly take their attributes where no class has gone before  Well, not if Mystic Champions – who are fully capable of stripping Cosmic Champions of their beneficial effects – have anything to say about it! Maybe it’s the Mystic Agenda to protect the secrets of the universe?
These changes ensure that having a Class Bonus always gives you the advantage it promises, as it now also reflects ability trends for a particular Class  Please keep in mind that these are generalizations, and some Champions abilities may not always strictly align with these relationships  Learn more about Champions’ abilities by viewing their profiles and tapping on features for detailed information 
• When you attack someone, you charge up their Power in addition to yours  This meant they would reach a full three bars while you only reached one and a half  We've reduced the amount defenders receive such that you'll be at two bars when they're at three  This change maintains the underdog functionality to give defenders a chance to comeback while being less punishing to players earning high Combos 
New damage types for attacks now play a larger role in the abilities of Champions  For example, some heroes power-up by successfully blocking magical damage, while others’ abilities may harm anyone that makes physical contact with them 
New Resistances and Immunities have found their way to the Battlerealm  Some heroes are completely immune to specific status effects based on either lore from the comics or logic  For example, the android Vision has no blood, and is therefore fully immune to Bleed conditions  We’ve also strengthened the effectiveness of certain status effects, so be careful who you choose to bring into battle! Could you guess who might be immune to the new “Poison” condition?
• Poison: Inflicts damage over time and reduces healing and regeneration effectiveness 
• Unstoppable: A buff to shrug off the impact from attacks, but still take the damage 
• Weakness: A debuff that reduces Attack attributes 
• Heal Block: Fully prevents the target from gaining health in any way 
• Power Lock: Seals the target, preventing them from gaining any Power 
• When fighting, you may notice that many status effects are now able to stack  This also changes how certain beneficial “buffs” and detrimental “debuffs” interact with one another  For example, it's now possible to have both Armor Up and Armor Break effects on you simultaneously  Let the tug-o-war begin, and may the strongest effects win!
• Black Bolt's Corkscrew: +25% damage, but at the cost of minor recoil damage 
• Punisher's “Wrath” has been replaced by "Payback"  Payback deals additional damage based on the total damage dealt to Frank 
• Colossus' “Unbreakable” now deal bonus damage based on his armor level at the time of activation 
• All of Black Panther’s special attacks now deal bonus damage based on the number of Bleeds on the target 
• Spider-Man’s Web-Slinger now has a chance to inflict Weakness 
• Vision’s Physical Disruption: Added a minor Power Burn effect due to “his” use of his Infrared Beam  “He” also now purges all status effects while phasing through the ground 
• Scarlet Witch: Increased the Critical Hit Chance for Hex Bolt and Hex Sphere 
• Many knockback effects have been adjusted to improve consistency 
We’ve tested the Signature Abilities quite extensively before releasing them, but there have been a few abilities that we have been keeping an eye on  We’ve compared our notes with the feedback you’ve been sending us and are making some balance changes to them  Thanks for your feedback!
Slightly reduced the frequency and duration of Juggernaut’s “Unstoppable” ability 
• He was indeed a bit too   unstoppable  We’ve toned down the frequency this ability triggers, as well as reduced the duration it’s active for when it does trigger  We feel Juggernaut is still a powerful Champion despite these revisions  Take care!
Slightly reduced the starting values of Wolverine's “Cellular Regeneration” 
• We found that Cellular Regeneration was too strong at lower levels where fewer counters to Regeneration exist 
Re-scaled Gamora's “Assassination” to start higher but scale slower 
• At lower levels, Special Attacks were used too infrequently, giving this powerful ability little visibility  We’ve adjusted the scaling to better match Special Attack usage at all levels 
Increased the frequency that Black Bolt’s “Provocation” triggers 
• Due to the varying Critical Hit rates across all Champions, in some cases Provocation would trigger rarely or not at all within a fight  We’ve increased the frequency to ensure you’ll see it every match – but especially so against opponents with high Critical Hit rates 
We’ll continue to follow the effect of these new abilities on gameplay  Please keep your feedback coming!
Hey everyone! We have been hard at work on improving the game and have prepared a big update inspired in part by your great community feedback  Please keep letting us know what you think!
• Fixed many Dash, Medium, Heavy and Special Attacks missing or failing to execute 
• Added Alliances and a new Alliance Crystal 
• Rocket Raccoon and Unstoppable Colossus join The Contest 
• Temporary Boosts to Attack, Health, and XP are now available from the Alliance Crystal 
• Rewards for completing and exploring Chapters and Acts  Earn a guaranteed 3-Star hero crystal for each fully explored Act! This is retroactive, just complete any quest to claim them 
• A new Fight Menu combines The Arenas, Story Quests and Event Quest menus 
• Updated Summoner Profiles with new information  Inspect other players’ Profiles and brag about your achievements!
• A list of blocked users has been added to Chat windows  The option to unblock these users is found in this new menu  The power is in your hands now!
• We fixed Dash and Medium Attack issues for many heroes that sometimes missed or did not activate 
• We fixed issues to Drax and Colossus Light and Medium Attacks where they would not connect 
• Fixed an issue where the camera would stop moving after a level 3 special sequence 
• Fixed an issue where the player’s heavy attack would get stuck in charge even after the player has released input 
• Fixed a rare bug where Champions were still able to deal damage after they died, resulting in tied fights 
Form Alliances with your Friends!
What is better than playing? Playing with your friends! Create a new Alliance or join an existing one through the new Alliance Menu 
• Invite other players to your Alliance 
• Search for an Alliance by name or join a Recommended Alliance 
• Receive rewards for entering your first Alliance 
• Alliance News Feed  The news feed celebrates your Alliance member’s achievements 
• Alliance Chat  Chat with other members of your Alliance in a private channel all to yourself 
• Help Allies  Players can ask for help when out of Energy or Stamina  Alliance members help each other as much as they can to earn Loyalty Points  Loyalty points have a daily limit to how many can be earned 
• Alliance Crystal  Access a new Alliance Crystal while part of any Alliance  Use new Loyalty Points for purchasing Alliance Crystals 
He may start out slow, but watch out for his immense power at high ranks!
• Adjusted the range of many Heavy Attacks, including Hulk and Drax, to ensure they correctly connect with enemies 
• Many Special Attacks, including those for Wolverine, Iron Fist, Winter Soldier, Punisher, Black Panther, and many others have had their range adjusted to ensure they correctly connect with enemies even if activated immediately after a combo that knocked the enemy back 
• Payback and Unbreakable now display their maximum potential damage bonus 
• Added detailed descriptions for Bleed Immunity and Poison Immunity 
• Gamora: We’ve adjusted the scaling of her base Special Attack damage to ensure they scale up more similarly to other heroes  This also makes Gamora more reliant on her high Bleed damage, and improves the chances of opponents able to deal with her high Bleed 
Vital Strike and Jade Assassin damage decreased by 10% 
Godslayer damage increased by 10% 
• Magik: Rewind is a game-changer for Magik that allows her to go up against foes like Gamora and Rewind off big Critical Hits and Bleed damage; however, the frequency of Rewind triggering was too low to be there when she needed it 
Increased the likelihood Rewind triggers by +20% at all levels 
Rewind now heals over one second instead of instantly 
Fixed a bug allowing Magik to break out of an enemy combo using Rewind  It now only removes Status Effects 
• Hulk: Given the riskiness of losing Health in certain game modes, Hulk’s anger-management provided too little help too late in the game  We’ve increased the Attack boost to ensure he’s appropriately scary in all game modes – as long as he’s angry!
Increased Hulk Rage by +20% Attack at all ability levels 
Arc Overload no longer causes Armor Break when it expires 
• Vision: Added Poison Immunity to our robot friend 
Arena tuning is an ongoing process  The team is continually making adjustments to Arenas to improve the experience 
Ultron has infected The Contest!
Many new Champions join the battle against Ultron 
Quest through the new Ultron’s Assault Event 
Wield new power with Summoner Masteries 
Grow your Friend’s List with the new Social Hub 
Team up with your Alliance in new Events, Arenas, and more!
Filter and sort your Stash 
Fights have been optimized for performance improvements on all devices 
Users can now filter through the items in their Stash 
Fixed several issues where Hero Rating would fluctuate 
Fixed a bug with Rhino and Juggernaut having 11-20% more Armor than intended 
Fixed a bug with Rocket Raccoon’s Dash attack being slower than intended 
Added a confirmation popup when spending Units on stamina recharges and unlocking arenas 
Regeneration no longer displays green Health values if you’re at full Health 
Several new improvements to how status effects are displayed 
AI opponents are no longer able to perform one unavoidable attack in response to a Special Attack 3 
A new and improved look for all Health Potions in the Battlerealm 
All Revive Potions now revive your Champions with +10% more Health 
We’re adding so many new Champions, they could form their own Alliance!
Some of your favourite heroes of the Marvel Cinematic Universe join The Contest!
Summoner Mastery is on the horizon!
Masteries provide beneficial effects for your Champions 
Access Masteries through your Summoner Profile 
Earn Mastery Points when you level up 
Choose your Masteries wisely and strategically customize your benefits 
Recover your points to try a new specialization as often as you’d like 
Keep an eye on in-game messaging for more information 
The daily loyalty limit has been set to refresh at 08:00UTC for all players 
A timer has been added to show when the daily loyalty limit resets 
Loyalty balance is now displayed in the Alliance menus 
Ask for Versus help with a single tap on the ‘Help’ icon in Team Select 
New Alliance Events are coming very soon!
Work together with your Alliance to complete objectives and receive rewards!
Muster your might, Alliance Arenas will soon open their gates!
Competing in Alliance Arenas shares your points across your whole Alliance; work together to reach milestones and top ranks!
Work together to amass a huge score, and defeat your competition in classic Arena combat! No slackers here either - if you don’t contribute to win the competition, you’re not eligible for the goods!
All social features (Chat, Mail, and Friends) can now be accessed through the new Social Hub 
Search for and add friends, and send private messages to Summoners on your Friends List 
Redesigned chat and mail screens 
Take on other Summoners’ top Champions for bragging rights and prizes in 1-on-1 Duels!
A new series of special Ultron quests are available, starting with the first Chapter  Fight back against Ultron’s infection alongside the Summoner, and team up with some of Marvel’s finest! New quests unlock each week!
The Spider-Man Champion gate has been removed from Act 1, Chapter 1, Quest 5 
• Fixed an issue where chat snapped to the most recent message 
• Fixed several issues where Hero Rating would fluctuate 
• Various improvements to the Summoner Mastery screens and descriptions 
• Increased the ISO8 awarded by duplicate 2-Star Champions 
Quest through the new single-player campaign, Ant-Man’s Adventure!
In addition to Ant-Man and Yellowjacket feuding throughout the Battlerealm, additional new Champions will be joining The Contest!
Access more Masteries in the new Utility Mastery tree!
Please note, these changes may result in a loss of Hero Rating as incorrect effects are restored back to normal levels 
Improved and polished combat mechanics to reduce the amount of stutters and lost input 
Fixed and optimized rendering related issues with Metal enabled devices 
Team up with Ant-Man, and put a stop to Yellowjacket’s mysterious mission!
All Alliance Quests only last for a specified amount of time, defeat the boss with your Alliance before it expires!
New Prestige System - A dynamic difficulty and score setting that adjusts as you and your Alliance succeed in harder quests  The better you do and the tougher your Alliance is, the higher the prestige  The higher the prestige, the better the rewards!
Choose your teams carefully as Champions within Alliance Quests cannot be used in other Story or Event Quests 
Act 4 has been released! Play Chapter 1 now!
Summoner level maximum has been increased to level 60!
5-Star Champions are coΩming to The Contest! These are the most powerful Champions yet!
Additional improvements have been made to the UI, Versus Arenas, Synergy Bonuses, the Stash & Items Store 
Act 4 - Chapter 1 released!
New challenges - more path variation and features to challenge the strongest Summoners!
Greater challenge means greater rewards! Earn 4 Star Crystals and Mastery Points!
The Summoner Level cap has been increased by ten levels to level 60!
Champion Items will be coming soon! These allow you to apply items and buffs to a specific Champion, keep an eye out for updates on these new Champion Items!
Synergy Bonuses have updated iconography and the calculation has been updated to a distinct, additive bonus - What you see is what you get!
Alliance class distribution is now displayed on team select - Choose the right class!
Your Catalysts now have their own inventory, and will no longer appear in the Upgrade Item inventory 
The Stash is now separated into three tabs: Catalysts, Rewards and ISO, allowing you to sort and view your Stash much faster!
The UI flow for both Quests and Arenas have been greatly improved  You can now skip through fight victory and reward animations!
Here is the rundown of patch 5 1 0, filled with various bug fixes and optimizations  The important ones to note are below 
New Champions, new theme, and a new arena!
To celebrate our one year anniversary AND the holidays, we’ll be running a special event quest! Battle through the history of The Contest, and test your mettle against familiar faces both old and new!
A special reward will be available to those who master every quest!
Our Anniversary Celebration will be happening very soon; stay tuned for more info!
More Act 4 quests are coming very soon!
Opponents in Story Quests now have the ability to use their Special 3 attack! Note that we are not changing previous quest opponents to have this special attack (Act 1-3, Proving Grounds, Realm of Legends will not change); this will be in effect starting with the soon-to-be-released Act 4 content 
As with our previous major build releases (3 0’s Ultron, 4 0’s Ant Man, and 5 0’s Battlerealm), the Contest has been reskinned with a new theme!
The Road to Knowhere map is here! Fight in a new level inspired by Guardians of the Galaxy!
A new button in your Alliance Chat to take you directly to Alliance Quests!
You can now collect Catalyst Fragments in Event Quests, Proving Grounds, and Alliance Quests; these can be pieced together into a Catalyst!
Selling Items is now a thing! Sell any items in your inventory for gold!
Level 3 and Level 4 Health Potions have arrived! These are powerful instruments to help you tackle all the new Act 4 content!
Over 400 bugs were fixed in this patch!
This patch is a fix for the missing Champions during the Special 3 animation on Android devices 
This issue occurred during our upload process to the Google Play Store  This was an odd edge case scenario that we could not have caught during our internal tests, as it began appearing once we uploaded to the Google Play Store  This hotfix will be out by tomorrow, and will put Android at version 6 0 1  As this issue does not occur on iOS devices, iOS will remain at version 6 0 
3:30pm PST: We have started slow-rolling this patch out to Android devices, beginning with about 20% of users  We expect this to be available for 100% of users within the next 24 hours 
We have a few new Champions that you will see within the next couple of months (including one of my personal favorites)!
Over 200 total bugs squashed in this patch!
An artifact left over from the early days of the contest was Black Panther’s ability to gain a Critical Hit Rate boost during Special 3 attacks  As many might know, Critical Hits aren’t possible during a Special 3 anymore, making this effect   unhelpful  We’ve switched it out with a new ability to stack up even more Bleed effects on the opponent based on how many Bleeds are already active 
Example: The opponent has 4 stacks (instances) of Bleed on them when you launch a Special 3  With this new ability, you have a chance to add an additional 0 - 4 more stacks (instances) of Bleed 
Previously, a bug existed that allowed champions with Evade to continue to dodge Black Widow’s attacks, even if her Signature Ability was maxed out  This issue has been fixed 
Captain America WW2 has started to become outpaced by his non-WW2 counterpart and while we want the two to feel different and each have their own specific uses, we also want to ensure they are kept within range of each other in terms of balance  To accomplish this, we’ve given WW2 Cap the ability to Stun with his Special 1 and Special 3 attacks, but kept his Bleed on Special 2 the same, giving him options during combat against non-bleeding champions 
A bug that prevented Daredevil from triggering Armor Breaks from Heavy Attacks has been fixed and is now working as intended 
Against non-bleeding champions: Critical Hits have a chance to Armor Break on Special Attacks 
Increase range of Signature to 25% from 20% 
Many players found Elektra’s signature ability lacked enough opportunities to use it  To remedy this, we’ve increased the range from 20% to 25%  Additionally, to help make Elektra unique from other skill champions, we’ve given her the ability to deal with naturally Bleed Immune champions  Note: This Armor Break only applies to champions naturally immune to bleed, such as Colossus and Ultron, and not to champions granted Bleed Immunity from Local or Link Nodes 
Guillotine’s Bleed effect used to have a chance to activate from any given attack, meaning that it had to be kept quite weak to compensate for the frequency of triggers  We’ve made the switch to have her Bleed behave closer to existing champions, and in doing so have boosted the strength of the Bleed and have allowed it to stack 
Norman Osborn overloads the Arc Reactor in his chest if Health drops below 10%, granting a large burst of power, with (18% - 48% ) Armor, Regeneration, and Power Gain  After that, his suit burns out and cannot trigger Armor Up, Armor Break or Stun and loses all base Armor 
Many players didn’t like Iron Patriot’s old signature ability, feeling that due to the lack of Regeneration, it was considerably weaker than Iron Man’s  While we agreed, we didn’t want to just copy and paste his signature ability, but rather give him his own unique twist on the ability  This “all or nothing version” feels more like Norman Osborn, pushing his suit to the limit to get a larger boost but at the cost of damaging the suit  The addition of Power Gain allows Iron Patriot a large attack before the suit burns out, if timed correctly 
Heavy Attacks: 90% chance to Stagger the enemy for 8 seconds  A Staggered enemy cannot gain their next beneficial effect 
All versions of Juggernaut, even those who haven’t been awakened, now gain the 2 second Unstoppable ability at the start of the fight when they hit Rank 2 
We wanted to add some new functionality to Juggernaut, while also keeping him true to his Mystic class assignment  To accomplish this, we added this “buff smasher” effect which keeps an opponent from gaining their next beneficial effect  Additionally, we wanted to make non-awakened versions of Juggernaut more fun to play, without adding more power to the awakened variations  As a result, we gave all versions of Juggernaut the ability to become Unstoppable at the start of the fight 
While many players liked the new functionality of Star-Lord’s Element Gun effect, they found it to be a little too random, specifically when it would Heal Block a champion incapable of Healing  We’ve now added in some contingencies that will make Heal Block appear less unless the opposing champion shows that he / she can Heal during the fight  This includes both activated healing effects, such as Wolverine or Ultron’s Heal, or passive healing effects gained from Masteries, such as Salve or Willpower 
It’s been a bit weird that Bucky wasn’t friends with his most famous friend  Well, he is now  This affects 3 Star and above versions 
We’ve increased the overall speed of this attack, allowing quick players to use this ability after a four or five hit combo 
It seems the Marvel’s have gotten tired of their beams being dodged so easily and have decided to angle it a little better, increasing the overall range of the attack and making it harder to dodge away from  We’ve also increased the speed of both special attacks to allow them to better flow into combat 
In order to allow this attack to better flow in combat, we’ve shaved off a few frames from the beginning, allowing players to chain this attack into 4 and 5 hit combos 
Alliance Wars have arrived! It’s Alliance versus Alliance in a war for Battlerealm supremacy!
Enter the NEW Loyalty Store to buy Alliance Potions, Mastery items, or other EXCLUSIVE items 
Gain Power back from Special Attacks, enhance or defend against Special Attacks, OR gain a temporary Arena Point Boosts with hoards of new Summoner Boost items!
Additional changes and improvements are listed below 
This patch will be released February 24th 
A new area of the Battlerealm has been opened! Compete with your Alliance-mates for pride, glory, and PRIZES!
Matchmake to find a rival Alliance, then combine strategy and teamwork to dominate them 
Setup the ultimate defensive team to fortify your Battlerealm, then take your offensive team on the assault!
Watch your War Rating skyrocket as your Alliance works together to defeat rivals!
Load up on Crystal Shards, Loyalty, and brand new exclusive rewards!
Note that this will be slow-rolled to Alliances in phases, similar to the introduction of Alliance Quests (to ensure server stability and gather your feedback on the new mode)  Expect tuning changes throughout these phases, as well as into Season 1 
Use Loyalty instead of Units to obtain items for Alliance Quests & Wars!
Items will rotate daily, similar to how the Mastery cores in the current Store change 
Store contents will be randomly chosen from a pool of categories/items; a select few items will be persistent and always be available for purchase 
A 5-Star version of Unstoppable Colossus will be available in the Loyalty Store (keep in mind, this is an expensive Champion due to his exclusivity; this will require winning quite a few Alliance Wars and saving up!) 
This is accessible from the “Store” section of the pop-down menu, and will be available at a later date after the initial 7 0 launch; there will be advance notice through forums and in-game before we release the Loyalty Store 
New Summoner Boosts have arrived in the Loyalty Store; NEW Boost types, purchasable with Loyalty Points 
Class specific Boosts, such as Mystic Champions restoring power after using Special Attacks 2 and 3, or Skill Champions boosting their Special Attack Damage 
Defensive Boosts, where your Champions take reduced incoming Special 3 Attack Damage 
Gain a temporary Arena Point boost with new Arena Boost items!
Fixed an issue where, after Parrying certain Champion’s Special Attacks, your Champion would be stuck in a blocking state until the Special Attack finished 
Fixed an issue where 90s Cyclops’ Armor Breaks would not remove Armor Ups 
Fixed an issue with Scarlet Witch’s Signature Ability proc rate (previously, the % chance displayed did not match in-game functionality; this is now fixed) 
(Netflix) Daredevil’s Heavy Attack now has a chance to apply 2 stacks of Armor Break, instead of the previous 1 stack 
When spending Battlechips to enter an Arena (such as the Tier 4 Basic or Alpha Catalyst Arena), there is now a confirmation popup 
The Alliance Crystal now has a purchase limit that resets daily 
Permanently increased the Alliance Crystal’s points in Summoner Advancement (from 30 to 300) 
Updates to Champion Special Attack animations, flow, and timing 
7 0 1 will be released within the next few days 
A celebration message is sent to the War Room when an Alliance War battlegroup is cleared 
Players can now tap directly on another node icon while the tile info popup is open (previously, the popup had to be closed before selecting another node) 
Alliance’s reward tier position is now highlighted in the Alliance War tier breakdown 
In Attack Phase, players can view the score breakdown for both the battlegroup and overall 
The “Place Your Defenders” text now disappears much faster after tapping on the screen 
Mail messages now display the date they were sent 
It should be much harder to accidentally tap the Units Store when closing a screen 
Players can tap to skip the point animation in Versus mode again 
Resolved an issue with Class Masteries (specifically Mystic Dispersion) not functioning 
The Juggernaut issue with his linked nodes not appearing in Act 4, Chapter 3, Quest 3 (4 3 3) has been fixed 
Fixed a crash that occurs when a player who is not in an Alliance enters Alliance Wars through an outside link 
Fixed a text issue where Alliance War specific descriptions would appear on the Alliance Quest “Select a Battlegroup” screen 
Resolved ~20 various rare crashes and additional minor issues in different game modes 
Fixed and optimized performance on the new Samsung S7 
Fixed an Unknown Error that occurred rarely after a device was woken after going to sleep 
Improved Performance(Frames Per Second) tracking per fight to help diagnose hitches/pauses/lag spikes during gameplay 
Improved gesture tracking(Swipe, Tap, Hold) during low performance moments in combat 
Fixed a rare crash that would sometimes occur when receiving a phone call while in combat 
Tuned and updated many Champion Special Attack animations to improve timing and combat flow  Please see the expanded forum post HERE for a full list 
Fixed She-Hulk’s Special Attacks being marked as a projectile (allowing Daredevil to evade them) 
Fixed an issue where the player would be stuck in place after parrying Captain America’s Special 1 
Fixed an issue where chaining 2 medium attacks into Old Man Logan’s Special 2 would cause the first 2 strikes to miss opponents 
Fixed an issue with Daredevil or Spider-man missing with a dash attack if Vision charges a heavy attack during the dash 
Fixed an issue where some hidden information in Alliance Wars was visible 
Fixed a display issue where Defender Placement percentage was not displaying all placed Alliance members 
Resolved minor issue with the total Alliance’s score being displayed on the War Progress widget (now only displays the score of the battlegroup being viewed) 
Multiple minor Alliance War issues have also been fixed in this patch 
Fixed a display issue where Shard amounts provided by defeating a boss displayed as double 
Fixed a display issue where opponent PI values would display differently between the map, prefight screen, and in combat 
Boss power is now correctly displayed after removing Global and Linked boosts 
Fixed an issue where a player in Alliance Quests would lose input ability on the quest board after sleeping the device 
Fixed an issue where a player enters Alliance Quests and gets stuck after viewing the linked node or buff node tutorial 
Fixed an issue where sending an Alliance invite to a player would cause the “Add Friend” button to become greyed out 
Fixed a text issue that appears when viewing Featured Hero information from the Home Screen 
Join The Iron or fight for The Blue with new events, quests, Champions, and special Shards; inspired by Marvel’s Captain America: Civil War!
Solo Events: constantly-evolving events that vary in length, requirements, and prizes!
Compare statistics against other players and Alliances with the new Leaderboards!

Context 2: Paper Info

Title: Environmental variability and network structure determine the optimal plasticity mechanisms in embodied agents
Publish Date: Unkown
Author List: Sina Khajehabdollahi (from Department of Computer Science, University of Tübingen)

Figure

Figure2: An outline of the network controlling the foraging agent The sensor layer receives inputs at each time step (the ingredients of the nearest food), which are processed by the plastic layer in the same way as the static sensory network, Fig 1 The output of that network is given as input to the motor network, along with the distance d and angle α to the nearest food, the current velocity v, and energy E of the agent These signals are processed through two hidden layers to the final output of motor commands as the linear and angular acceleration of the agent
Figure4: The evolved parameters θ = (θ 1 ,      , θ 8 ) of the plasticity rule for the reward prediction (a ) and the decision (b ) tasks, for a variety of parameters (p tr = 0 01, d e ∈ 0, 0 1,      , 1, and σ ∈ 0, 0 1,      , 1 in all 100 combinations) Despite the relatively small difference between the tasks, the evolved learning rules differ considerably For visual guidance, the lines connect θs from the same run 
Figure5: a The trajectory of an agent (blue line) in the 2D environment A well-trained agent will approach and consume food with positive values (green dots) and avoid negative food (red dots) b The learning rate of the plastic sensory network eta p grows with the distance between environments d e c  and decreases with the frequency of environmental change d The fitness of an agent (measured as the total food consumed over its lifetime) increases over generations of the EA for both the scalar and binary readouts in the sensory network e The Pearson correlation coefficient of an evolved agent's weights with the ingredient value vector of the current environment (E 1 -blue, E 2 -red) In this example, the agent's weights are anti-correlated with its environment, which is not an issue for performance since the motor network can interpret the inverted signs of food 

abstract

The evolutionary balance between innate and learned behaviors is highly intricate, and different organisms have found different solutions to this problem  We hypothesize that the emergence and exact form of learning behaviors is naturally connected with the statistics of environmental fluctuations and tasks an organism needs to solve 
Here, we study how different aspects of simulated environments shape an evolved synaptic plasticity rule in static and moving artificial agents  We demonstrate that environmental fluctuation and uncertainty control the reliance of artificial organisms on plasticity  Interestingly, the form of the emerging plasticity rule is additionally determined by the details of the task the artificial organisms are aiming to solve 
Moreover, we show that coevolution between static connectivity and interacting plasticity mechanisms in distinct sub-networks changes the function and form of the emerging plasticity rules in embodied agents performing a foraging task  One of the defining features of living organisms is their ability to adapt to their environment and incorporate new information to modify their behavior 
It is unclear how the ability to learn first evolved , but its utility appears evident  Natural environments are too complex for all the necessary information to be hardcoded genetically and more importantly, they keep changing during an organism's lifetime in ways that cannot be anticipated ;   The link between learning and environmental uncertainty and fluctuation has been extensively demonstrated in both natural ; , and artificial environments  
Nevertheless, the ability to learn does not come without costs  For the capacity to learn to be beneficial in evolutionary terms, a costly nurturing period is often required, a phenomenon observed in both biological , and artificial organisms   Additionally, it has been shown that in some complex environments, hardcoded behaviors may be superior to learned ones given limits in the agent's lifetime and envi-ronmental uncertainty ; ;  
The theoretical investigation of the optimal balance between learned and innate behaviors in natural and artificial systems goes back several decades  However, it has recently found also a wide range of applications in applied AI systems ;   Most AI systems are trained for specific tasks, and have no need for modification after their training has been completed 
Still, technological advances and the necessity to solve broad families of tasks make discussions about life-like AI systems relevant to a wide range of potential application areas  Thus the idea of open-ended AI agents that can continually interact with and adapt to changing environments has become particularly appealing 
Many different approaches for introducing lifelong learning in artificial agents have been proposed  Some of them draw direct inspiration from actual biological systems ;   Among them, the most biologically plausible solution is to equip artificial neural networks with some local neural plasticity , similar to the large variety of synaptic plasticity mechanisms ; ; that performs the bulk of the learning in the brains of living organisms  
The artificial plasticity mechanisms can be optimized to modify the connectivity of the artificial neural networks toward solving a particular task  The optimization can use a variety of approaches, most commonly evolutionary computation  The idea of meta-learning or optimizing synaptic plasticity rules to perform specific functions has been recently established as an engineering tool that can compete with stateof-the-art machine learning algorithms on various complex tasks ; ; Pedersen and Risi (2021);  
Additionally, it can be used to reverse engineer actual plasticity mechanisms found in biological neural networks and uncover their functions ;   Here, we study the effect that different factors (environ-arXiv:2303 06734v1 [q-bio NC] 12 Mar 2023 mental fluctuation and reliability, task complexity) have on the form of evolved functional reward-modulated plasticity rules 
We investigate the evolution of plasticity rules in static, single-layer simple networks  Then we increase the complexity by switching to moving agents performing a complex foraging task  In both cases, we study the impact of different environmental parameters on the form of the evolved plasticity mechanisms and the interaction of learned and static network connectivity 
Interestingly, we find that different environmental conditions and different combinations of static and plastic connectivity have a very large impact on the resulting plasticity rules  We imagine an agent who must forage to survive in an environment presenting various types of complex food particles  Each food particle is composed of various amounts and combinations of N ingredients that can have positive (food) or negative (poison) values 
The value of a food particle is a weighted sum of its ingredients  To predict the reward value of a given resource, the agent must learn the values of these ingredients by interacting with the environment  The priors could be generated by genetic memory, but the exact values are subject to change  To introduce environmental variability, we stochastically change the values of the ingredients 
More precisely, we define two ingredient-value distributions E 1 and E 2 and switch between them, with probability p tr for every time step  We control how (dis)similar the environments are by parametrically setting E 2 = (1 − 2d e )E 1 , with d e ∈ [0, 1] serving as a distance proxy for the environments; when d e = 0, the environment remains unchanged, and when d e = 1 the value of each ingredient fully reverses when the environmental transition happens 
For simplicity, we take values of the ingredients in E 1 equally spaced between -1 and 1 (for the visualization, see Fig  )  The static agent receives passively presented food as a vector of ingredients and can assess its compound value using the linear summation of its sensors with the (learned or evolved) weights, see Fig   
The network consists of N sensory neurons that are projecting to a single post-synaptic neuron  At each time step, an input X t = (x 1 ,       , x N ) is presented, were the value x i , i ∈ {1,       , N } represents the quantity of the ingredient i  We draw x i independently form a uniform distribution on the [0, 1] interval (x i ∼ U (0, 1)) 
The value of each ingredient w c i is determined by the environment (E 1 or E 2 )  The postsynaptic neuron outputs a prediction of the food X t value as y t = g(W X T t )  Throughout the paper, g will be either the identity function, in which case the prediction neuron is linear, or a step-function; however, it could be any other nonlinearity, such as a sigmoid or ReLU 
After outputting the prediction, the neuron receives feedback in the form of the real value of the input R t   The real value is computed as R t = W c X T t + ξ, where W c = (w c 1 ,       , w c N ) is the actual value of the ingredients, and ξ is a term summarizing the noise of reward and sensing system ξ ∼ N (0, σ) 
Figure : An outline of the static agent's network  The sensor layer receives inputs representing the quantity of each ingredient of a given food at each time step  The agent computes the prediction of the food's value y t and is then given the true value R t ; it finally uses this information in the plasticity rule to update the weight matrix 
For the evolutionary adjustment of the agent's parameters, the loss of the static agent is the sum of the mean squared errors (MSE) between its prediction y t and the reward R t over the lifetime of the agent  The agent's initial weights are set to the average of the two ingredient value distributions, which is the optimal initial value for the case of symmetric switching of environments that we consider here 
As a next step, we incorporate the sensory network of static agents into embodied agents that can move around in an environment scattered with food  To this end, we merge the static agent's network with a second, non-plastic motor network that is responsible for controlling the motion of the agent in the environment 
Specifically, the original plastic network now provides the agent with information about the value of the nearest food  The embodied agent has additional sensors for the distance from the nearest food, the angle between the current velocity and the nearest food direction, its own velocity, and its own energy level (sum of consumed food values) 
These inputs are processed by two hidden layers (of 30 and 15 neurons) with tanh activation  The network's outputs are angular and linear acceleration, Fig    The embodied agents spawn in a 2D space with periodic boundary conditions along with a number of food particles that are selected such that the mean of the food value distribution is ∼ 0  An agent can eat food by approaching it sufficiently closely, and each time a food particle is eaten, it is The sensor layer receives inputs at each time step (the ingredients of the nearest food), which are processed by the plastic layer in the same way as the static sensory network, Fig   
The output of that network is given as input to the motor network, along with the distance d and angle α to the nearest food, the current velocity v, and energy E of the agent  These signals are processed through two hidden layers to the final output of motor commands as the linear and angular acceleration of the agent re-spawned with the same value somewhere randomly on the grid (following the setup of ) 
After 5000 time steps, the cumulative reward of the agent (the sum of the values of all the food it consumed) is taken as its fitness  During the evolutionary optimization, the parameters for both the motor network (connections) and plastic network (learning rule parameters) are co-evolved, and so agents must simultaneously learn to move and discriminate good/bad food 
Reward-modulated plasticity is one of the most promising explanations for biological credit assignment   In our network, the plasticity rule that updates the weights of the linear sensor network is a rewardmodulated rule which is parameterized as a linear combination of the input, the output, and the reward at each time step:
Additionally, after each plasticity step, the weights are normalized by mean subtraction, an important step for the stabilization of Hebbian-like plasticity rules   We use a genetic algorithm to optimize the learning rate η p and amplitudes of different terms θ = (θ 1 ,       , θ 8 )  The successful plasticity rule after many food presentations must converge to a weight vector that predicts the correct food values (or allows the agent to correctly decide whether to eat a food or avoid it) 
To have comparable results, we divide θ = (θ 1 ,       , θ 8 ) by We then multiply the learning rate η p with θ max to maintain the rule's evolved form unchanged, η norm p = η p • θ max   In the following, we always use normalized η p and θ, omitting norm   To evolve the plasticity rule and the moving agents' motor networks, we use a simple genetic algorithm with elitism  
The agents' parameters are initialized at random (drawn from a Gaussian distribution), then the sensory network is trained by the plasticity rule and finally, the agents are evaluated  After each generation, the bestperforming agents (top 10 % of the population size) are selected and copied into the next generation 
The remaining 90 % of the generation is repopulated with mutated copies of the best-performing agents  We mutate agents by adding independent Gaussian noise (σ = 0 1) to its parameters  To start with, we consider a static agent whose goal is to identify the value of presented food correctly  The static reward-prediction network quickly evolves the parameters of the learning rule, successfully solving the prediction task 
We first look at the evolved learning rate η p , which determines how fast (if at all) the network's weight vector is updated during the lifetime of the agents  We identify three factors that control the learning rate parameter the EA converges to: the distance between the environments, the noisiness of the reward, and the rate of environmental transition 
The first natural factor is the distance d e between the two environments, with a larger distance requiring a higher learning rate, Fig    This is an expected result since the convergence time to the "correct" weights is highly dependent on the initial conditions  If an agent is born at a point very close to optimality, which naturally happens if the environments are similar, the distance it needs to traverse on the fitness landscape is small 
Therefore it can afford to have a small learning rate, which leads to a more stable convergence and is not affected by noise
Context 3: Paper Info

Title: Efficient nonparametric estimation of Toeplitz covariance matrices
Publish Date: March 20, 2023
Author List: Karolina Klockmann (from Department of Statistics and Operations Research, Universität Wien), Tatyana Krivobokova (from Department of Statistics and Operations Research, Universität Wien)

Figure

Figure 1: Spectral density functions (first row) and autocovariance functions (second row) for examples 1, 2, 3 
Figure 2: Distance between the first atom and the first center of mass of aquaporin (left) and the opening diameter y t over time t (right) 
black line in the left plot) confirms that the covariance matrix estimated with our VST-DCT method almost completely decorrelates the channel diameter Y on the training data set Next, we estimated the regression coefficients β with the usual PLS algorithm, ignoring the dependence in the data Finally, we estimated β with PLS that takes into account dependence using our covariance estimator Σ Based on these regression coefficient estimators, the prediction on the test set was calculated The plot on the right side of Figure 2 shows the Pearson correlation between the true channel diameter on the test set and the prediction on the same test set based on raw (grey) and decorrelated data (black) 
Figure 3: On the left, the auto-correlation function of Y (grey) and of Σ−1/2 Y (black), where Σ is estimated with the VST-DCT method; On the right, correlation between the true values on the test data set and prediction based on partial least squares (in grey) and corrected partial least squares (black) 
Uniform distributionThe observations follow a uniform distribution with covariance matrices Σ 1 , Σ 2 , Σ 3 of examples 1, 2, 3, i e , Y i = Σ 1/2 j X i , j = 1, 3, with X 1 ,    the parameter innov of the R function arima sim is used to pass the innovations X 1 ,    , X n i i d Table4, 5 and 6 show respectively the results for (A) p = 5000, n = 1, (B) p = 1000, n = 50 and (C) p = 5000, n = 10 
(A) p = 5000, n = 1: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral and L 2 norm, respectively, as well as the average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(C) p = 5000, n = 10: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral and L 2 norm, respectively, as well as the average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(A) p = 5000, n = 1: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(B) p = 1000, n = 50: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(C) p = 5000, n = 10: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 

abstract

A new nonparametric estimator for Toeplitz covariance matrices is proposed  This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression  The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data-driven and computationally very fast 
Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices  These results are readily extended to estimation of inverses of Toeplitz covariance matrices  Also, an alternative version of the Whittle likelihood for the spectral density based on the Discrete Cosine Transform (DCT) is proposed 
The method is implemented in the R package vstdct that accompanies the paper 

Introduction

Estimation of covariance and precision matrices is a fundamental problem in statistical data analysis with countless applications in the natural and social sciences  Covariance matrices with a Toeplitz structure arise in the study of stationary stochastic n = 1, to the best of our knowledge, there is no fully data-driven approach for selecting the banding/tapering/thresholding parameter 
suggested first to split the time series into non-overlapping subseries and then apply the cross-validation criterion of   However, it turns out that the right choice of the subseries length is crucial for this approach, but there is no data-based method available for this  In this work, an alternative way to estimate a Toeplitz covariance matrix and its inverse is chosen 
Our approach exploits the one-to-one correspondence between Toeplitz covariance matrices and their spectral densities  First, the given data are transformed into approximate Gaussian random variables whose mean equals to the logarithm of the spectral density  Then, the log-spectral density is estimated by a periodic smoothing spline with a data-driven smoothing parameter 
Finally, the resulting spectral density estimator is transformed into an estimator for Σ or its inverse  It is shown that this procedure leads to an estimator that is fully data-driven, automatically positive definite and achieves the minimax optimal convergence rate under the spectral norm over a large class of Toeplitz covariance matrices 
In particular, this class includes Toeplitz covariance matrices that correspond to long-memory processes with bounded spectral densities  Moreover, the computation is very efficient, does not require iterative or resampling schemes and allows to apply any inference and adaptive estimation procedures developed in the context of nonparametric Gaussian regression 
Estimation of the spectral density from a stationary time series is a research topic with a long history  Earlier nonparametric methods are based on smoothing of the (log-)periodogram, which itself is not a consistent estimator   Another line of nonparametric methods for estimating the spectral density is based on the Whittle likelihood, which is an ap-proximation to the exact likelihood of the time series in the frequency domain 
For example, estimated the spectral density from a penalized Whittle likelihood, while used polynomial splines to estimate the log-spectral density function maximizing the Whittle likelihood  Recently, Bayesian methods for spectral density estimation have been proposed (see , but these may become very computationally intensive in large samples due to posterior sampling 
The minimax optimal convergence rate for nonparametric estimators of Hölder continuous spectral densities from Gaussian stationary time series was obtained by under the L p , 1 ≤ p ≤ ∞, norm  Only few works on spectral density estimation show the optimality of the corresponding estimators  In particular, and derived convergence rates of their estimators for the log-spectral density under the L 2 norm, while neglecting the Whittle likelihood approximation error 
In general, most works on spectral density estimation do not exploit further the close connection to the corresponding Toeplitz covariance matrix estimation  In particular, an upper bound for the L ∞ risk of a spectral density estimator automatically provides an upper bound for the risk of the corresponding Toeplitz covariance matrix estimator under the spectral norm 
This fact is used to establish the minimax optimality of our nonparametric estimator for the Toeplitz covariance matrices  The main contribution of this work is to show that our proposed spectral density estimator is not only numerically very efficient, but also achieves the minimax optimal rate in the L ∞ norm, which in turn ensures the minimax optimality of the corresponding Toeplitz covariance matrix estimator 
The paper is structured as follows  In Section 2, the model is introduced and ap-proximate diagonalization of Toeplitz covariance matrices with the discrete cosine transform is discussed  Moreover, an alternative version of the Whittle's likelihood is proposed  In Section 3, new estimators for the Toeplitz covariance matrix and the precision matrix are derived, while in Section 4 their theoretical properties are presented 
Section 5 contains simulation results, Section 6 presents a real data example, and Section 7 closes the paper with a discussion  The proofs are given in the appendix to the paper 

Set up and diagonalization of Toeplitz matrices

Let Y 1 ,       , Y n i i d  ∼ N p (0 p , Σ), where Σ is a (p × p)-dimensional positive definite covariance matrix with a Toeplitz structure, that is, Σ = {σ |i−j| } p i,j=1 0  The sample size n may tend to infinity or to be a constant  The case n = 1 corresponds to a single observation of a stationary time series, and in this case the data are simply denoted by Y ∼N p (0 p , Σ) 
The dimension p is assumed to grow  The spectral density function f , corresponding to a Toeplitz covariance matrix Σ, is given by so that for f ∈ L 2 (−π, π) the inverse Fourier transform implies Hence, Σ is completely characterized by f , and the non-negativity of the spectral density function implies the positive definiteness of the covariance matrix 
Moreover, the decay of the autocovariance σ k is directly connected to the smoothness of f   Finally, the convergence rate of a Toeplitz covariance estimator and that of the corresponding spectral density estimator are directly related via Σ ≤ f ∞ := sup x∈ |f (x)|, where • denotes the spectral norm (see  
As in , we introduce a class of positive definite Toeplitz covariance matrices with Hölder continuous spectral densities  For β = γ + α > 0, where The optimal convergence rate for estimating Toeplitz covariance matrices over P β (M 0 , M 1 ) depends crucially on β  It is well known that the k-th Fourier coefficient of a function whose γ-th derivative is α-Hölder continuous decays at least with order O(k −β ) (see  
Hence, β determines the decay rate of the autocovariances σ k , which are the Fourier coefficients of the spectral density f , as k → ∞  In particular, this implies that for β ∈ (0, 1], the class P β (M 0 , M 1 ) includes Toeplitz covariance matrices corresponding to long-memory processes with bounded spectral densities, since the sequence of corresponding autocovariances is not summable 
A connection between Toeplitz covariance matrices and their spectral densities is further exploited in the following lemma  Lemma 1  Let Σ ∈ P β (M 0 , M 1 ) and let x j = (j − 1)/(p − 1), j = 1,    , p, then where δ i,j is the Kroneker delta, O(•) terms are uniform over i, j = 1,       , p and divided by √ 2 when i, j ∈ {1, p} is the Discrete Cosine Transform I (DCT-I) matrix 
The proof can be found in Appendix A 1  This result shows that the DCT-I matrix approximately diagonalizes Toeplitz covariance matrices and that the diagonalization error depends to some extent on the smoothness of the corresponding spectral density  In the spectral density literature the discrete Fourier transform (DFT) matrix
, where i is the imaginary unit, is typically employed to approximately diagonalize Toeplitz covariance matrices  Using the fact that introduced an approximation for the likelihood of a single Gaussian stationary time series (case n = 1), the so-called Whittle likelihood (1) The quantity , where F j denotes the j-th column of F , is known as the periodogram at the j-th Fourier frequency 
Note that due to periodogram symmetry, only p/2 data points I 1 ,    , I p/2 are available for estimating the mean f (2πj/p), j = 1,       , p/2 , where x denotes the largest integer strictly smaller than x  The Whittle likelihood has become a popular tool for parameter estimation of stationary time series, e g , for nonparametric and parametric spectral density estimation or for estimation of the Hurst exponent, see e g , ;  
Lemma 1 yields the following alternative version of the Whittle likelihood where W j = (D t j Y ) 2   Note that this likelihood approximation is based on twice as many data points W j as the standard Whittle likelihood  Thus, it allows for a more efficient use of the data Y to estimate the parameter of interest, such as the spectral density or the Hurst parameter 
Equations ( ) or (2) invite for the estimation of f by maximizing the (penalized) likelihood over certain linear spaces (e g , spline spaces), as suggested e g , in or   However, such an approach requires well-designed numerical methods to solve the corresponding optimization problem, since the spectral density in the second term of (1) or ( ) is in the denominator, which does not allow to obtain a closed-form expression for the estimator and often leads to numerical instabilities 
Also, the choice of the smoothing parameter becomes challenging  Therefore, we suggest an alternative approach that allows the spectral density to be estimated as a mean in an approximate Gaussian regression  Such estimators have a closed-form expression, do not require an iterative optimization algorithm and a smoothing parameter can be easily obtained with any conventional criterion 
First Hence, for W j = (D t j Y ) 2 , j = 1,       , p it follows with Lemma 1 that where Γ(a, b) denotes a gamma distribution with a shape parameter a and a scale parameter b  Note that the random variables W 1 ,       , W p are only asymptotically independent  Obviously, E(W j ) = f (πx j ) + O(1), j = 1,      
, p  To estimate f from W 1 ,       , W p , one could use a generalized nonparametric regression framework with a gamma distributed response, see e g , the classical monograph by   However, this approach requires an iterative procedure for estimation, e g , a Newton-Raphson algorithm, with a suitable choice for the smoothing parameter at each iteration step 
Deriving the L ∞ rate for the resulting estimator is also not a trivial task  Instead, we suggest to employ a variance stabilizing transform of that converts the gamma regression into an approximate Gaussian regression  In the next section we present the methodology in more detail for a general setting with n ≥ 1 

Methodology

For Y i ∼ N p (0 p , Σ), i = 1,       , n, it was shown in the previous section that with Lemma 1 the data can be transformed into gamma distributed random variables     , n, j = 1,       , p, where for each fixed i the random variable W i,j has the same distribution as W j given in (3)  Now the approach of Cai et al  ( ) is adapted to the setting n ≥ 1 
First, the transformed data points W i,j are binned, that is, fewer new variables     , T   Note that the number of observations in a bin is m = np/T   In Theorem 1 in Section 4, we show that setting T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) leads to the minimax optimal rate for the spectral density estimator 
To simplify the notation, m is handled as an integer (otherwise, one can discard several observations in the last bin)  Next, applying the variance stabilizing transform (VST) ∼ where H(y) = {φ(m/2) + log (2y/m)} / √ 2 and φ is the digamma function (see   Now, the scaled and shifted log-spectral density H(f ) can be estimated with a periodic smoothing spline
where h > 0 denotes a smoothing parameter, q ∈ N is the penalty order and S per (2q − 1) a space of periodic splines of degree 2q − 1  The smoothing parameter h can be chosen either with generalized cross-validation (GCV) as derived in or with the restricted maximum likelihood, see   Once an estimator H(f ) is obtained, application of the inverse transform function H −1 (y) = m exp √ 2y − φ (m/2) /2 yields the spectral density estimator f = H −1 H(f )  
Finally, using the inverse Fourier transform leads to the fol- The precision matrix Ω is estimated by the inverse Fourier transform of the reciprocal of the spectral density estimator, i e , Ω = (ω |i−j| ) p i,j=1 with ωk = The estimation procedure for Σ and Ω can be summarised as follows  1  Data Transformation:
where D is the (p × p)-dimensional DCT-I matrix as given in Lemma 1 and D j is its j-th column  2  Binning: Set T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) and calculate W i,j , k = 1,       , T 

VST:

where k are asymptotically i i d  Gaussian variables  Inverse VST: Estimate the spectral density f with f = H −1 H(f ) , where Note that Σ and Ω are positive definite matrices by construction, since their spectral density functions f and f −1 are non-negative, respectively  Unlike the banding and tapering estimators, the autocovariance estimators σk are controlled by a single smoothing parameter h, which can be estimated fully data-driven with several available automatic methods, which are numerically efficient and well-studied 
In addition, one can also use methods for adaptive mean estimation, see e g , , which in turn leads to adaptive Toeplitz covariance matrix estimation  All inferential procedures developed in the Gaussian regression context can also be adopted accordingly 

Theoretical Properties

In this section, we study the asymptotic properties of the estimators f , Σ and Ω  The results are established under the asymptotic scenario where p → ∞ and p/n → c ∈ (0, ∞], that is, the dimension p grows, while the sample size n either remains fixed or also grows but not faster than p  This corresponds to the asymptotic scenario when the sample covariance matrix is inconsistent 
Let f be the spectral density estimator defined in Section 3, i e , f = m exp{ √ 2 H(f ) − φ(m/2)}/2, where H(f ) is given in (4), m = np/T and φ is the digamma function  Furthermore, let Σ be the Toeplitz covariance matrix estimator and Ω the corresponding precision matrix defined in equations ( ) and (6), respectively 
The following theorem shows that both Σ and Ω attain the minimax optimal rate of convergence over the class and hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) and q = max{1, γ}, the spectral density estimator f , the corresponding covariance matrix estimator Σ and the precision matrix estimator Ω satisfy sup
For h {log(np)/(np)}   The proof of Theorem 1 can be found in the Appendix A 3 and is the main result of our work  The most important part of this proof is the derivation of the convergence rate for the spectral density estimator f under the L ∞ norm  In the original work, established an L 2 rate for a wavelet nonparametric mean estimator in a gamma regression where the data are assumed to be independent 
In our work, the spectral density estimator f is based on the gamma distributed data W i,1 ,       , W i,p , which are only asymptotically independent  Moreover, the mean of these data is not exactly f (πx 1 ),       , f (πx p ), but is corrupted by the diagonalization error given in Lemma 1  This error adds to the error that arises via binning and VST and that describes the deviation from a Gaussian distribution, as derived in  
Finally, we need to obtain an L ∞ rather than an L 2 rate for our spectral density estimator  Overall, the proof requires different tools than those used in   To get the L ∞ rate for f , we first derive that for the periodic smoothing spline estimator H(f ) of the log-spectral density  To do so, we use a closed-form expression of its effective kernel obtained in , thereby carefully treating various (dependent) errors that describe deviations from a Gaussian nonparametric regression with independent errors and mean f (πx i ) 
Note also that although the periodic smoothing spline estimator is obtained on T binned points, the rate is given in terms of the vector dimension p  Then, using the Cauchy-Schwarz inequality and a mean value argument, this rate is translated into the L ∞ rate for the spectral density estimator f   To obtain the rate for the Toeplitz covariance matrix estimator is enough to note that

Simulation Study

In this section, we compare the performance of the proposed Toeplitz covariance estimator, denoted as VST-DCT, with the tapering estimator of and with the sample covariance matrix  We consider Gaussian vectors Y 1 ,    , (3) such that the corresponding spectral density is Lipschitz continuous but not differentiable: f (x) = 1 44{| sin(x + 0 5π)| 1 7 + 0 45} 
In particular, var(Y i ) = 1 44 in all three examples  Figure shows the spectral densities and the corresponding autocorrelation functions for the three examples  A Monte Carlo simulation with 100 iterations is performed using R (version 4 1 2, seed 42)  For our VST-DCT estimator, we use a cubic periodic spline, i e , q = 2 is set in (4) 
The binning parameters are set to T = 500 bins with m = 10 points for (A) and T = 500 bins with m = 100 points for both (B) and (C)  To select the regularisation parameter for our estimator, we implemented the restricted maximum likelihood (ML) method, generalized cross validation (GCV) and the corresponding oracle versions, i e , as if Σ were known The tapering parameter
where T ap k (Σ ν 2 ) is the tapering estimator of with parameter k  If n = 1, that is, under scenario (A), suggest to split the time series Y into l non-overlapping subseries of length p/l and then proceed as before to select the tuning parameter k  To the best of our knowledge, there is no data-driven method for selecting this parameter l 
Using the true covariance matrix Σ, we selected l = 30 subseries for the example 1 and l = 15 subseries for the exam-ples 2 and 3  The parameter k can then be chosen by cross-validation as above  We employ this approach under scenario (A) instead of an unavailable fully data-driven criterion and name it semi-oracle 
Finally, for all three scenarios (A), (B) and (C), the oracle tapering parameter is computed using grid search for each Monte Carlo sample as kor = arg min k=2,3,   ,p/2 T ap k ( Σ) − Σ , where Σ is the sample covariance matrix  To speed up the computation, one can replace the spectral norm by the 1 norm, as suggested by  
In Tables , the errors of the Toeplitz covariance estimators with respect to the spectral norm and the computation time for one Monte Carlo iteration are given for scenarios (A), (B) and (C), respectively  To illustrate the goodness-of-fit of the spectral density, the L 2 norm f − f 2 is also computed 
The results show that the tapering and VST-DCT estimator perform overall similar in terms of the spectral norm risk  This is not surprising as both estimators are proved to be rate-optimal  Moreover, both the tapering and VST-DCT estimators are clearly superior to the inconsistent sample Toeplitz covariance matrix 
A closer look at the numbers shows that the VST-DCT method has better constants, i e , VST-DCT estimators have somewhat smaller errors in the spectral norm than the tapering estimators across all examples, but especially under scenario (C)  The oracle estimators show similar behaviour, but are slightly less variable compared to the data-driven estimators 
In general, both the tapering and VST-DCT estimators perform best for example 1, second best for example 3 and worst for example 2, which traces back to functions complexity  In terms of computational time, both methods are similarly fast for scenarios (A) and (B)  For scenario (C), the tapering method is much slower due to the multiple high-dimensional matrix multiplications in the cross-validation method 
It is expected that for larger p the tapering estimator is much more computationally intensive than the corresponding VST-DCT estimator  (1) polynomial σ ( (  To test how robust our approach is to deviations from the Gaussian assumption, we simulated the data from gamma and uniform distributions and conducted a simulation study for the same scenarios and examples 
The results are very similar to those of the Gaussian distribution, see supplementary materials for the details 

Application to Protein Dynamics

We revisit the data analysis of protein dynamics performed in Krivobokova et al  (2012) and 



ground_truth: Players can skip dialogue on the quest map by pressing the 'SKIP' button.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the transition probability of the environment affect the learning rate in the static agent?

Read the following text and answer briefly.

Context 2:  A second parameter that impacts the learning rate is the variance of the rewards  The reward an agent receives for the plasticity step contains a noise term ξ that is drawn from a zero mean Gaussian distribution with standard deviation σ 
This parameter controls the unreliability of the agent's sensory system, i e , higher σ means that the information the agent gets about the value of the foods it consumes cannot be fully trusted to reflect the actual value of the foods  As σ increases, the learning rate η p decreases, which means that the more unreliable an environment becomes, the less an agent relies on plasticity to update its weights, Fig   
Indeed for some combinations of relatively small distance d e and high reward variance σ, the EA converges to a learning rate of η p ≈ 0  This means that the agent opts to have no adaptation during its lifetime and remain at the mean of the two environments  It is an optimal solution when the expected loss due to ignoring the environmental transitions is, on average, lower than the loss the plastic network will incur by learning via the (often misleading because of the high σ) environmental cues 
A final factor that affects the learning rate the EA will converge to is the frequency of environmental change during an agent's lifetime  Since the environmental change is modeled as a simple, two-state Markov process (Fig  ), the control parameter is the transition probability p tr   When keeping everything else the same, the learning rate rapidly rises as we increase the transition probability from 0, and after reaching a peak, it begins to decline slowly, eventually reaching zero (Fig  ) 
This means that when environmental transition is very rare, agents opt for a very low learning rate, allowing a slow and stable convergence to an environment-appropriate weight vector that leads to very low losses while the agent remains in that environment  As the rate of environmental transition increases, faster learning is required to speed up convergence in order to exploit the (comparatively shorter) stays in each environment 
Finally, as the environmental transition becomes too fast, the agents opt for slower or even no learning, which keeps them ) and the decision (b ) tasks, for a variety of parameters (p tr = 0 01, d e ∈ 0, 0 1,       , 1, and σ ∈ 0, 0 1,       , 1 in all 100 combinations)  Despite the relatively small difference between the tasks, the evolved learning rules differ considerably 
For visual guidance, the lines connect θs from the same run  near the middle of the two environments, ensuring that the average loss of the two environments is minimal (Fig  )  The form of the evolved learning rule depends on the task: Decision vs  Prediction The plasticity parameters θ = (θ 1 ,       , θ 8 ) for the rewardprediction task converge on approximately the same point, regardless of the environmental parameters (Fig  ) 
In particular, θ 3 → 1, θ 5 → −1, θ i → 0 for all other i, and thus the learning rule converges to: Since by definition y t = g(W t X T t ) = W t X T t (g(x) = x in this experiment) and R t = W c X T t + ξ we get: Thus the distribution of ∆W t converges to a distribution with mean 0 and variance depending on η p and σ and W converges to W c  
So this learning rule will match the agent's weight vector with the vector of ingredient values in the environment  We examine the robustness of the learning rule the EA discovers by considering a slight modification of our task  Instead of predicting the expected food value, the agent now needs to decide whether to eat the presented food or not 
This is done by introducing a step-function nonlinearity (g(x) = 1 if x ≥ 1 and 0 otherwise)  Then the output y(t) is computed as: Instead of the MSE loss between prediction and actual value, the fitness of the agent is now defined as the sum of the food values it chose to consume (by giving y t = 1)  Besides these two changes, the setup of the experiments remains exactly the same 
The qualitative relation between η p and parameters of environment d e , σ and p tr is preserved in the changed experiment  However, the resulting learning rule is significantly different (Fig  )  The evolution converges to the following learning rule: In both cases, the rule has the form ∆W t = η p X t [α y R t + β y ] 
Thus, the ∆W t is positive or negative depending on whether the reward R t is above or below a threshold (γ = −β y /α y ) that depends on the output decision of the network (y t = 0 or 1)  Both learning rules (for the reward-prediction and decision tasks) have a clear Hebbian form (coordination of preand post-synaptic activity) and use the incoming reward signal as a threshold 
These similarities indicate some common organizing principles of reward-modulated learning rules, but their significant differences highlight the sensitivity of the optimization process to task details  We now turn to the moving embodied agents in the 2D environment  To optimize these agents, both the motor network's connections and the sensory network's plasticity parameters evolve simultaneously 
Since the motor network is initially random and the agent has to move to find food, the number of interactions an agent experiences in its lifetime can be small, slowing down the learning  However, having the larger motor network also has benefits for evolution because it allows the output of the plastic network to be read out and transformed in different ways, resulting in a broad set of solutions 
The fitness of an agent (measured as the total food consumed over its lifetime) increases over generations of the EA for both the scalar and binary readouts in the sensory network  e  The Pearson correlation coefficient of an evolved agent's weights with the ingredient value vector of the current environment (E 1 -blue, E 2 -red) 
In this example, the agent's weights are anti-correlated with its environment, which is not an issue for performance since the motor network can interpret the inverted signs of food  The agents can solve the task effectively by evolving a functional motor network and a plasticity rule that converges to interpretable weights (Fig  ) 
After ∼ 100 evolutionary steps (Fig  ), the agents can learn the ingredient value distribution using the plastic network and reliably move towards foods with positive values while avoiding the ones with negative values  We compare the dependence of the moving and the static agents on the parameters of the environment: d e and the state transition probability p tr  
At first, in order to simplify the experiment, we set the transition probability to 0, but fixed the initial weights to be the average of E 1 and E 2 , while the real state is E 2   In this experiment, the distance between states d e indicates twice the distance between the agent's initial weights and the optimal weights (the environment's ingredient values) since the agent is initialized at the mean of the two environment distributions 
Same as for the static agent, the learning rate increases with the distance d e (Fig  )  Then, we examine the effect of the environmental transition probability p tr on the evolved learning rate η p   In order for an agent to get sufficient exposure to each environment, we scale down the probability p tr from the equivalent experiment for the static agents 
We find that as the probability of transition increases, the evolved learning rate η p decreases (Fig  )  This fits with the larger trend for the static agent, although there is a clear difference when it comes to the increase for very small transition probabil-ities that were clearly identifiable in the static but not the moving agents 
This could be due to much sparser data and possibly the insufficiently long lifetime of the moving agent (the necessity of scaling makes direct comparisons difficult)  Nevertheless, overall we see that the associations observed in the static agents between environmental distance d e and transition probability p tr and the evolved learning rate η p are largely maintained in the moving agents 
Still, more data would be needed to make any conclusive assertions about the exact effect of these environmental parameters on the emerging plasticity mechanisms  A crucial difference between the static and the moving agents is the function the plasticity has to perform  While in the static agents, the plasticity has to effectively identify the exact value distribution of the environment in order to produce accurate predictions, in the embodied agents, the plasticity has to merely produce a representation of the environment that the motor network can evolve to interpret adequately enough to make decisions about which food to consume 
To illustrate the difference, we plot the Pearson correlation coefficient between an agent's weights and the ingredient values of the environment it is moving in (Fig  )  We use the correlation instead of the MSE loss (which we used for the static agents in Fig  ) because the amplitude of the vector varies a lot for different agents and meaningful The evolved parameters of moving agents' plasticity rule for the g(s) = x, identity (a ) and the step function (Eq 
4) (b ) sensory networks (the environmental parameters here are d e ∈ [0, 1], σ = 0 and p tr = 0 001)  The step function (binary output) network evolved a more structured plasticity rule (e g , θ 3 > 0 for all realizations) than the linear network  Moreover, the learned weights for the identity network (c ) have higher variance and correlate significantly less with the environment's ingredient distribution compared to the learned weights for the thresholded network (d )
conclusions cannot be drawn from the MSE loss  For many agents, the learned weights are consistently anti-correlated with the actual ingredient values (an example of such an agent is shown in Fig  )  This means that the output of the sensory network will have the opposite sign from the actual food value 
While in the static network, this would lead to very bad predictions and high loss, in the foraging task, these agents perform exactly as well as the ones where the weights and ingredients values are positively correlated, since the motor network can simply learn to move towards food for which it gets a negative instead of a positive sensory input 
This additional step of the output of the plastic network going through the motor network before producing any behavior has a strong effect on the plasticity rules that the embodied agents evolve  Specifically, if we look at the emerging rules the top performing agents have evolved (Fig  ), it becomes clear that, unlike the very well-structured rules of the static agents (Fig  ), there is now virtually no discernible pattern or structure 
The difference becomes even clearer if we look at the learned weights (at the end of a simulation) of the best-performing agents (Fig  )  While there is some correlation with the environment's ingredient value distribution, the variance is very large, and they do not seem to converge on the "correct" values in any way 
This is to some extent expected since, unlike the static agents where the network's output has to be exactly correct, driving the evolution of rules that converge to the precise environmental distribution, in the embodied networks, the bulk of the processing is done by the motor network which can evolve to interpret the scalar value of the sensory network's output in a variety of ways 
Thus, as long as the sensory network's plasticity rule co-evolves with the motor network, any plasticity rule that learns to produce consistent information about the value of encountered food can potentially be selected  To further test this assumption, we introduce a bottleneck of information propagation between the sensory and motor networks by using a step-function nonlinearity on the output of the sensory network (Eq 
4)  Similarly to the decision task of the static network, the output of the sensory network now becomes binary  This effectively reduces the flow of information from the sensory to the motor network, forcing the sensory network to consistently decide whether food should be consumed (with the caveat that the motor network can still interpret the binary sign in either of two ways, either consuming food marked with 1 or the ones marked with 0 by the sensory network) 
The agents perform equally well in this variation of the task as before (Fig  ), but now, the evolved plasticity rules seem to be more structured (Fig  )  Moreover, the variance of the learned weights in the bestperforming agents is significantly reduced (Fig  ), which indicates that the bottleneck in the sensory network is in-creasing selection pressure for rules that learn the environment's food distribution accurately 
We find that different sources of variability have a strong impact on the extent to which evolving agents will develop neuronal plasticity mechanisms for adapting to their environment  A diverse environment, a reliable sensory system, and a rate of environmental change that is neither too large nor too small are necessary conditions for an agent to be able to effectively adapt via synaptic plasticity 
Additionally, we find that minor variations of the task an agent has to solve or the parametrization of the network can give rise to significantly different plasticity rules  Our results partially extend to embodied artificial agents performing a foraging task  We show that environmental variability also pushes the development of plasticity in such agents 
Still, in contrast to the static agents, we find that the interaction of a static motor network with a plastic sensory network gives rise to a much greater variety of wellfunctioning learning rules  We propose a potential cause of this degeneracy; as the relatively complex motor network is allowed to read out and process the outputs from the plastic network, any consistent information coming out of these outputs can be potentially interpreted in a behaviorally useful way 
Reducing the information the motor network can extract from the sensory system significantly limits learning rule variability  Our findings on the effect of environmental variability concur with the findings of previous studies that have identified the constraints that environmental variability places on the evolutionary viability of learning behaviors 
We extend these findings in a mechanistic model which uses a biologically plausible learning mechanism (synaptic plasticity)  We show how a simple evolutionary algorithm can optimize the different parameters of a simple reward-modulated plasticity rule for solving simple prediction and decision tasks 
Reward-modulated plasticity has been extensively studied as a plausible mechanism for credit assignment in the brain ; ; and has found several applications in artificial intelligence and robotics tasks ;   Here, we demonstrate how such rules can be very well-tuned to take into account different environmental parameters and produce optimal behavior in simple systems 
Additionally, we demonstrate how the co-evolution of plasticity and static functional connectivity in different subnetworks fundamentally changes the evolutionary pressures on the resulting plasticity rules, allowing for greater diversity in the form of the learning rule and the resulting learned connectivity 
Several studies have demonstrated how, in biological networks, synaptic plasticity heavily interacts with and is driven by network topology   Moreover, it has been recently demonstrated that biological plasticity mechanisms are highly redundant in the sense that any observed neural connectivity or recorded activity can be achieved with a variety of distinct, unrelated learning rules  
This observed redundancy of learning rules in biological settings complements our results and suggests that the function of plasticity rules cannot be studied independently of the connectivity and topology of the networks they are acting on  The optimization of functional plasticity in neural networks is a promising research direction both as a means to understand biological learning processes and as a tool for building more autonomous artificial systems 
Our results suggest that reward-modulated plasticity is highly adaptable to different environments and can be incorporated into larger systems that solve complex tasks  This work studies a simplified toy model of neural network learning in stochastic environments  Future work could be built on this basic framework to examine more complex reward distributions and sources of environmental variability 
Moreover, a greater degree of biological realism could be added by studying more plausible network architectures (multiple plastic layers, recurrent and feedback connections) and more sophisticated plasticity rule parametrizations  Additionally, our foraging simulations were constrained by limited computational resources and were far from exhaustive 
Further experiments can investigate environments with different constraints, food distributions, multiple seasons, more complex motor control systems and interactions of those systems with different sensory networks as well as the inclusion of plasticity on the motor parts of the artificial organisms 



ground_truth: As the transition probability increases, the learning rate initially rises and then declines.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the advantage of decorrelating the data before running the PLS algorithm?

Read the following text and answer briefly.

Context 3:  We consider data generated by the molecular dynamics (MD) simulations for the yeast aquaporin (Aqy1) -the gated water channel of the yeast Pichi pastoris  MD simulations are an established tool for studying biological systems at the atomic level on timescales of nano-to microseconds 
The data are given as Euclidean coordinates of all 783 atoms of Aqy1 observed in a 100 nanosecond time frame, split into 20 000 equidistant observations  Additionally, the diameter of the channel y t at time t is given, measured by the distance between two centers of mass of certain residues of the protein 
The aim of the analysis is to identify the collective motions of the atoms responsible for the channel opening  In order to model the response variable y t , which is a distance, based on the motions of the protein atoms, we chose to represent the protein structure by distances between atoms and certain fixed base points instead of Euclidean coordinates 
That is, we calculated where A t,i ∈ R 3 , i = 1,       , 783 denotes the i-th atom of the protein at time t, B j ∈ R 3 , j = 1, 2, 3, 4, is the j-th base point and d(•, •) is the Euclidean distance  Figure shows the diameter y t and the distance between the first atom and the first center of mass  It can therefore be concluded that a linear model Y = Xβ + holds, where
  This linear model has two specific features which are intrinsic to the problem: first, the observations are not independent over time and second, X t is high-dimensional at each t and only few columns of X are relevant for Y   have shown that the partial least squares (PLS) algorithm performs exceptionally well on this type of data, leading to a small-dimensional and robust representation of proteins, which is able to identify the atomic dynamics relevant for Y  
Singer et al  ( ) studied the convergence rates of the PLS algorithm for dependent observations and showed that decorrelating the data before running the PLS algorithm improves its performance  Since Y is a linear combination of columns of X, it can be assumed that Y and all columns of X have the same correlation structure 
Hence, it is sufficient to estimate Σ = cov(Y ) to decorrelate the data for the PLS algorithm, i e , Σ −1/2 Y = Σ −1/2 Xβ + Σ −1/2 results in a standard linear regression with independent errors  Our goal now is to estimate Σ and compare the performance of the PLS algorithm on original and decorrelated data 
For this purpose, we divided the data set into a training and a test set (each with p = 10 000 observations)  First, we tested whether the data are stationary  The augmented Dickey-Fuller test confirmed stationarity for Y with a p-value< 0 01  The Hurst exponent of Y is 0 85, indicating moderate long-range dependence supported by a rather slow decay of the sample autocovariances (see grey line in the left plot of Figure ) 
Therefore, we set q = 1 for the VST-DCT estimator to match the low smoothness of the corresponding spectral density  Moreover, the smoothing parameter is selected with the restricted maximum likelihood method and T = 550 bins are used  Obviously, the performance of the PLS algorithm on the decorrelated data is significantly better for the small number of components 
In particular, with just one PLS component, the correlation between the true opening diameter on the test set and its prediction that takes into account the dependence in the data is already 0 54, while it is close to zero for PLS that ignores the dependence in the data  showed that the estimator of β based on one PLS component is exactly the ensemble-weighted maximally correlated mode (ewMCM), which is defined as the collective mode of atoms that has the highest probability to achieve a specific alteration of the response Y  
Therefore, an accurate estimator of this quantity is crucial for the interpretation of the results and can only be achieved if the dependence in the data is taken into account  Estimating Σ with a tapered covariance estimator has two practical problems  First, since we only have a single realization of a time series Y (n = 1), there is no datadriven method for selecting the tapering parameter 
Second, the tapering estimator turned out not to be positive definite for the data at hand  To solve the second problem, we truncated the corresponding spectral density estimator ftap to a small positive value, i e , f + tap = max{ ftap , 1/ log(p)} (see   To select the tapering parameter with cross-validation, we experimented with different subseries lengths and found that the tapering estimator is very sensitive to this choice 
For example, estimating the tapered covariance matrix based on subseries of length 8/15/30 yields a correlation of 0 42/0 53/0 34 between the true diameter and the first PLS component, respectively  Altogether, our proposed estimator is fully data-driven, fast even for large sample sizes, automatically positive definite and can handle certain long-memory processes 
In contrast, the tapering estimator is not data-driven and must be manipulated to become positive definite  Our method is implemented in the R package vstdct 

Discussion

In this paper, we proposed a simple, fast, fully data-driven, automatically positive definite and minimax optimal estimator of Toeplitz covariance matrices from a large class that also includes covariance matrices of certain long-memory processes  Our estimator is derived under the assumption that the data are Gaussian 
However, simulations show that the suggested approach yields robust estimators even when the data are not normally distributed  In the context of spectral density estima- , for mixing processes (see Theorem 5 3 of Rosenblatt, 2012), as well as for non-linear processes (see   Since DFT and DCT matrices are closely related, we expect that equation (3) also holds asymptotically for these non-Gaussian time series, but consider a rigorous analysis to be beyond the scope of this paper 
In fact, our numerical experiments have even shown that if the spectral density is estimated from W j = f (πx j ) + j , that is, as if W j were Gaussian instead of gamma distributed, then the resulting spectral density estimator has almost the same L ∞ risk (and hence the corresponding covariance matrix has almost the same spectral norm) 
Of course, such an estimator would lead to a wrong inference about f (πx j ), since the growing variance of W j would be ignored  Since our approach translates Toeplitz covariance matrix estimation into a mean estimation in an approximate Gaussian nonparametric regression, all approaches developed in the context of Gaussian nonparametric regression, such as (locally)
adaptive estimation, as well as the corresponding (simultaneous) inference, can be directly applied  Bayesian tools for adaptive estimation and inference in Gaussian nonparametric regression as proposed in can also be employed 

Appendix

Throughout the appendix, we denote by c, c 1 , C, C 1 ,       etc  generic constants, that are independent of n and p  To simplify the notation, the constants are sometimes skipped and we write for less than or equal to up to constants  We embed the p-dimensional Toeplitz matrix Σ = toep(σ 0 ,       , σ p−1 ) in a (2p − 2)dimensional circulant matrix Σ = toep(σ 0 ,      
, σ p−1 , σ p−2 ,       , σ 1 )  Then, Σ = with the conjugate transpose U * , and Λ is a diagonal matrix with the k-th diagonal value for k = 1,    , p given by Furthermore, Σ = V * ΛV , where V ∈ C (2p−2)×p contains the first p columns of U   In particular, b(j, r) = b(j, 2p−r) and c(j, r) = −c(j, 2p−r) for r = p+1,      
, 2p−2  Together, we have (A 1) Some calculations show that for r = 1,       p Using the Taylor expansion of cot(x) for 0 < |x| < π one obtains for r = 1,       p where the O term does not depend on j and the hidden constant does not depend on r, p  If i = j, equations (A 1) -(A 3) imply where the O terms do not depend on j 
Since the complex exponential function is Lipschitz continuous with constant L = 1, it holds λ r = λ j + L r,j |r − j|p −1 where −1 ≤ L r,j ≤ 1 is a constant depending on r, j  Then, , it is sufficient to consider j = 1,    , p − 1  We begin with first sum  For a shorter notation, we use k := r − 1 and l := j − 1 in the following 
Then, summing the squares of the first term in (A 4) for l = 0,    , p−2 on sums of reciprocal powers  If p is even, then the residual terms are given by where φ and φ (1) denote the digamma function and its derivative  If p is odd, similar remainder terms can be derived  To see that R i (l, p) = O(p −1 ) for i = 1, 2, 3 and uniformly in l we use that asymptotically φ(x)∼ log(x)−1/(2x) and
The mixed term are both of the order p −1   Furthermore, since the harmonic sum diverges at a rate of log(p)  Finally, λ j = f (x j )+O{log(p)p −β } by the uniform approximation properties of the discrete Fourier series for Hölder continuous functions (see   All together, we have shown that (DΣD) j,j = where the O terms are uniform over j = 1,    , p 
Case i = j and |i − j| is even In this case, (DΣD) i,j = a i a j uniformly in i, j  To show that a i a j 2p−2 r=1 λ r c(i, r)c(j, r) = O(p −1 ), we proceed similarly as before  Setting k=r−1, l=j−1, m=i−1 and using that l =m and |l−m| is even, one obtains where for even p the residual terms are given by If p is odd, analogous residual terms can be derived 
Using similar techniques as before, one can show that the two residual terms and the remaining mixed and square terms vanish at a rate of the order O(p −1 ) and uniformly in i, j  Case i = j and |i − j| is odd |r − i| and |r − j| are either odd and even, or even and odd  Without loss of generality, assume that |r − i| is even 
Then, (DΣD) i,j = a i a j 2p−2 r=1 λ r b(i, r)c(j, r)  Since b(i, •) is an even function, c(j, •) is an odd function and λ r = λ 2p−r , it follows (DΣD) i,j = 0  The structure of the proof is as follows  First, we derive the L ∞ rate of the periodic smoothing spline estimator H(f )  Then, using the Cauchy-Schwarz inequality and a mean value argument, the convergence rate of the spectral density estimator f is
∞ the first claim of the theorem follows  Finally, we prove the second statement on the precision matrices  For the sake of clarity, some technical lemmas used in the proof are listed separately in A 4  hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{1, β})/3, 1), the estimator H(f ) described in Section 3 with q = max{1, γ} satisfies
Proof : Application of the triangle inequality yields a bias-variance decomposition Set T = 2T − 2 and x k = (k − 1)/ T for k = 1,    , T   Using Lemma 4, we can write where Mirroring and renumerating ζ k , η k , k is similar as for Y * k , k = 1,    , T   Using the above representation, one can write First we reduce the supremum to a maximum over a finite number of points 
If q > 1, then W (•, x k ) is Lipschitz continuous with constant L > 0  In this case, it holds almost surely that sup ) is a piecewise linear function with knots at x j = j/ T   The factor (ζ k + ξ k ) can be considered as stochastic weights that do not affect the piecewise linear property  Thus, the supremum is attained at one of the knots x j = j/ T , j = 1,    , T , and (A 7) is also valid for q = 1 
Again with (a + b) 2 ≤ 2a 2 + 2b 2 we obtain We start with bounding   This requires a bound on 1 • ψ 2 denotes the sub-Gaussian norm  In case of a Gaussian random variable the norm equals to the variance  Thus with Lemma 2 and Lemma 4, we obtain Lemma 1 6 of ) then yields Recall that T = p υ for some fixed υ ∈ ((4 − 2 min{1, β})/3, 1) 
Using the inequality log(x) ≤ x a /a one can find constants x υ , C υ > 0 depending on υ but not on n, p such that log(2 T ) log(p) Next, we derive a bound for the second term The exponential decay property of the kernel K stated in Lemma 2 yields The first term in (A 9) can be bounded again with Lemma 1 6 of  
We use the fact that for not necessarily independent random variables X 1 ,    , X N R and R > 0 are constants  This is a consequence of Lemma 1 of which yields , it follows that N i=1 a i X i has a subGaussain distribution and the subGaussian norm is bounded by 2R( N i=1 a 2 i ) 1/2   See for further details on the subgaussian distribution 
T h   For the second inequality Lemma 2(ii) is used  Applying Lemma 1 6 of then yields To bound the second term in (A 9), we use the moment bounds for ξ k derived in Lemma 4  Then, for all integers > 1 Combining the error bounds (A 10) and (A 11) and choosing R=m −1/2 gives By assumption T = p υ and m = np (1−υ) for some fixed υ ∈ ((4 − 2 min{1, β})/3, 1) 
If is an integer such that ≥ 1/(1 − υ), then where we used log(x) ≤ x a /a with a = 1/(4 )  Consider 1/2 < β ≤ 1 and let 0 < χ < 1 be a constant  Applying log(x) ≤ x a /a twice with a = χ/(2 ) yields For any fixed υ∈((4 − 2 min{1, β})/3, 1) one can find an integer which is independent of n, p such that the right side of (A 12) holds 
Since p/n → c ∈ (0, ∞] and thus n/p = O(1) and p −1 = O(n −1 ), it follows for satisfying (A 12) that In total, choosing an integer Using the representation in Lemma 4 once more gives for each x ∈ [0, 1] The bounds on k in Lemma 4 imply Consider the case that β ≥ 1  In particular, q = γ and f (q) is α-Hölder continuous 
Since f is a periodic function with f (x) ∈ [δ, M 0 ] and H(y) ∝ φ(m/2)+ log (2y/m), it follows that {H(f )} (q) is also α-Hölder continuous  Extending g := H(f ) to the entire real line, we get Expanding g(t) in a Taylor series around x and using that h −1 K h is a kernel of order 2q, see Lemma 2(iii), it follows that for any x ∈ [0, 1]
where ξ x,t is a point between x and t  Using the fact that the kernel K h decays exponentially and that g (q) is α-Hölder continuous on [δ, M 0 ] with some constant L, the logarithm is Lipschitz continuous on a compact interval, it follows g = H(f ) is β-Hölder continuous  Expanding g to the entire line and using Lemma 2(iii) with
In a similar way as before, one obtains Note that T −β =o(h β ) as β > 1/2, T h → ∞ and h → 0 by assumption  Since the derived bounds are uniform for x ∈ [0, 1] it holds Putting the bounds A 13 and A 14 together gives If h > 0 such that h → 0 and hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{1, β})/3, 1), the estimator f described in Section 3 with q = max{1, γ} satisfies
Proof : By the mean value theorem, it holds for some function g between H(f ) and To show that the second term on the right hand side of (A 15) is negligible we use the moment generating function of H(f ) ∞   In the next paragraph, we derive the asymptotic order of E[exp{λ H(f ) ∞ }] for n, p → ∞, where λ > 0 may depend on n, p or not 
By the exponential decay property of the kernel K stated in Lemma 2 holds First, H(f ) ∞ is bounded with the maximum over a finite number of points  Calculating the derivative of s : Since δ δx s(x) > 0 almost surely for x = x k , the extrema occur at x k , k = 1,    , T   Thus, for λ > 0 the moment generating function of H(f ) ∞ is bounded by
Let M j = ( T h) −1 T k=1 γ h (x j , x k ), which by Lemma 2 is bounded uniformly in j by some global constant M > 0  By the convexity of the exponential function we obtain √ 2 and by assumption 0 ≤ δ ≤ f ≤ M 0   Using Lemma 3, Q k can be written as a sum of m = np/T independent gamma random variables, i e 
The moment generating function of | log(X)| when X follows a Γ(a, b)-distribution is given by where Γ(a) is the gamma function and γ(a, b) is the lower incomplete gamma function  In particular, To derive the asymptotic order of E[exp{λ H(f ) ∞ }] for n, p → ∞ we first establish the asymptotic order of the ratio Γ(a + t)/Γ(a) for a → ∞ 
We distinguish the two cases where t is independent of a and where t linearly depends on a  Thus, for 0 < t < a and t independent of a, equation (A 17) implies for a → ∞ that Γ(a + t)/Γ(a) = O(a t )  Similarly, it can be seen that Γ(a − t)/Γ(a) = O(a −t )  If 0 < t < a and t linearly depends on a, i e  t = ca for some constant c ∈ (0, 1), then we get Γ(a ± t)/Γ(a) = O(a ±t exp{a}) for a → ∞ 
Hence, for a fixed λ not depending on n, p and such that 0 < λ < m/( √ 2M j ) we get for sufficiently large n, p If λ = cm such that 0 < λ < m/( √ 2M j ), then for sufficiently large n, p b∈{cδ/m,cM 0 /m} (bm/2) for some constant L > 1  Set K = min j=1,   , T 1/( √ 2M j ) which is a constant independent of n, p  Altogether, we showed that for 0 < λ < Km and n, p → ∞
Bounding the right hand side of (A 15) for some constants c 0 , c 1 > 0 and n, p → ∞ Since g lies between H(f ) and H(f ), and f almost surely pointwise  Thus, for C > f ∞ = M 0 it holds where c 1 := H(C − M 0 )  Applying Markov inequality for t = cm with c ∈ (0, K) and C = 2L 4/c + M 0 where c, K, L are the constants in gives
Together with Proposition 1 follows Using the fact that the spectral norm of a Toeplitz matrix is upper bounded by the sup norm of its spectral density we get sup According to the mean value theorem, for a function g between H(f ) and H(f ), it holds that some constant c 1 > 0 not depending on n, p  Chosing the same constant C as in section A 3 2 it follows
Noting that 1/f ∞ ≤ 1/δ and 2/m exp {φ(m/2)} ∈ [0 25, 1] for m ≥ 1, (A 18) implies for some constants c 2 , c 3 > 0 and n, p → ∞ Since the derived bounds hold for each Σ(f ) ∈ F β , we get all together sup This section states some technical lemmata needed for the proof of Theorem 1  The proofs can be found in the supplementary material 
The first lemma lists some properties of the kernel K h and its extension K h on the real line  The proof is based on   Lemma 2  Let h > 0 be the bandwith parameter depending on N   (i) There are constants 0 < C < ∞ and 0 < γ < 1 such that for all for x, t ∈ [0, 1] Lemma 3 states that the sum of the correlated gamma random variables in each bin can be rewritten as a sum of independent gamma random variables 
for i = 1,    , n and j = (k − 1)m + 1,    , km, and x j = (j − 1)/(2p − 2)  Finally, Lemma 4 gives explicit bounds for the stochastic and deterministic errors of the variance stabilizing transform  Thus, it quantifies the difference to an exact Gaussian regression setting  This result is a generalization of Theorem 1 of Cai et al 
(2010) adapted to our setting with n ≥ 1 observations and correlated observations  √ 2 can be written as where for the proof of the first statement  Furthermore, for x, t ∈ [0, 1] holds In particular, for some constants C 1 , C 2 > 0 depending on γ ∈ (0, 1) but not on h and x, it holds h (iii) See Lemma 15 of with p = 2q − 1 
It is sufficient to show the statement for n = 1 by independence of the Y i   Then, the number of points per bin is m = p/T   For simplicity, the index i is skipped in the following  First, we write Q k as a matrix-vector product and refactor it so that it corresponds to a sum of independent scaled χ 2 random variables 
In the second step, we calculate the scaling factors  Let E (km) be a diagonal matrix with ones on the (k − 1)m + 1,    , km-th entries and otherwise zero diagonal elements  Then, By Theorem 1 of for the gamma distribution it follows where Wi,j iid  ∼ Γ(1/2, 2 f (x * k )) and such that Cov( Wi,j , Wi,h ) = Cov(W i,j , W i,h ) for j = (k − 1)p/T + 1,    , kp/T and h ∈ {1,    , p} \ {(k − 1)p/T + 1,    , kp/T } 
Let θ be the maximum difference of the observations' means in each bin  Then, θ = max are defined via quantile coupling, it holds Z k = Φ −1 {F Q( Qk )} (see   Furthermore, define the uniform random variables Let ρ = Cov(Z k , Z l )  Then, the identity implies F Z,Z (x, y) − Φ(x)Φ(y) ≥ 0 for all x, y ∈ R ⇐⇒ ρ ≥ 0, (see  
Since Cov( Qk , Ql ) ≥ 0 and the ratio of two densities is non-negative, x = − 2/m, it follows that f Q(x) is monotone decreasing for x ≥ − 2/m  Furthermore, F Q(− m/2) ≤ 0 5 for all m ∈ N as f Q(x) is right-skewed  In particular, − m/2 ≤ F −1 Q (1/2) for all m ∈ N  Finally, since f Q(− 2/m) → φ(0) for m → ∞ there is a constant c > 0 not depending on m such that
The simulation study in Section 5 is performed in the same way, but with the uniform and the gamma distribution instead of the Gaussian distribution 



ground_truth: Decorrelating the data before running the PLS algorithm improves the performance of the algorithm.